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► 



I 



PROPERTY OF 




Artes scie'ntia v«r«ta^ 






c,/'^ 



LOGIC OP CHANCE. 



[ 




4 



THE 



LOGHC OF CHANCE 



AN ESSAY 

ON THE FOUNDATIONS AND PROVINCE OP 
THE THEORY OF PROBABILITY, 

WITH ESPECIAL REFBBBNCE TO ITS APPLICATION TO 

MORAL AND SOCIAL SCIENCE. 



BT 

JOHN VENN, M.A. 

FBLLOW OF OOKVILLB AKD OAIUS OOLLBOB, OAHBBIDGK. 



"So careful of the tn>e she seems 
So careless of the single life.** 



SlottDon anO ®am(tit»3f: 
MACMILLAN AND CO. 

1866 



Camlbrttrge: 

FEINTED BY C. J. CLAY, MJi.. 
AT THE UNIVEKSITY PRESS. 



111 






M 



PREFACE. 



Any work on Probability by a Cambridge man 
will be so likely to have its scope and its general 
treatment of the subject prejudged, that it may 
be well to state at the outset that the following 
Essay is in no sense mathematical. Not only, to 
quote a common but often delusive assurance, will 
' no knowledge of mathematics beyond the simple 
rules of Arithmetic' be required to understand 
these pages, but it is not intended that any such 
knowledge should be acquired by the process of 
reading them. On the two or three occasions on 
which algebraical formulae occur they will not be 
found to form any essential part of the text. 

The science of ProbabiUty occupies at present 
a somewhat anomalous position. It is impossible, 
I think, not to observe in it some of the marks 
and consequent disadvantages of a secUanal study. 
By a small body of ardent students it has been 
cultivated with great assiduity, and the results 



VI PREFACE. 

they have obtained will always be reckoned 
among the most extraordinary products of mathe- 
matical genius. But by the general body of 
thinking men its principles seem to be regarded 
with indifference or suspicion. Such persons may 
admire the ingenuity displayed, and be struck 
with the profundity of many of the calculations, 
but there seems to them, if I may so express it, 
an umreality about the whole treatment of the 
subject. To many persons the mention of Proba- 
bility suggests little else than the notion of a set 
of rules, very ingenious and profound rules no 
doubt, with which mathematicians amuse them- 
selves by setting and solving puzzles. 

It must be admitted that some ground has 
been given for such an opinion. The examples 
commonly selected by writers on the subject, 
though very well adapted to illustrate its rules, 
are for the most part of a special and peculiar 
character, such as those relating to dice and 
cards. When they have searched for illustra- 
tions drawn from the practical business of life, 
they have generally, by a strange fatality, hit 
upon just the sort of instances which, as I shall 
endeavour to show hereafter, are among the very 
worst that could be chosen for the purpose. It is 
scarcely possible for any unprejudiced person to 
read what has been written about the credibility 



PEEFACB. TU 

of witnesses by eminent writers, without his ex- 
periencing an invincible distrust of the principles 
which they adopt. To say that the rules of evi- 
dence sometimes given by such writers are broken 
in practice, would scarcely be correct; for the rules 
are of a kind which generally defies any attempt 
to appeal to them in practice. 

This supposed want of harmony between Pro- 
bability and other branches of Philosophy is 
perfectly erroneous. It arises from the belief 
that Probability is a branch of mathematics trying 
to intrude itself on to ground which does not 
belong to it. I shall endeavour to show that this 
belief is unfounded. To answer correctly the sort 
of questions to which the science introduces us 
does generally demand some knowledge of mathe- 
matics^ often a great knowledge, but the discussion 
of the fundamental principles on which the rules 
are based does not necessarily require any such 
qualification. Questions might arise in other 
sciences, in Geology, for example, which could 
only be answered by the aid of arithmetical cal- 
culations. In such a case any one would admit 
that the arithmetic was extraneous and acci- 
dental However many questions of this kind 
there might be here, those persons who do not 
care to work out special results for themselves 
might still have an accurate knowledge of the 



VIU PBEFACE. 

principles of the science, and even considerable 
acquaintance with the details of it. The same 
holds true in Probability ; its connexion with ma- 
thematics, though certainly far closer than that of 
most other sciences, is still of much the same 
kind. It is principally when we wish to work out 
results for ourselves that mathematical knowledge 
is required; without such knowledge the student 
may still have a firm grasp of the principles and 
even see his way to many of the derivative re- 
sults. 

The opinion that Probability, instead of being 
a branch of the general science of evidence which 
happens to make much use of mathematics, is a 
portion of mathematics, erroneous as it is, has yet 
been very disadvantageous to the science in several 
ways. Students of Philosophy in general have 
thence conceived a prejudice against Probability, 
which has for the most part deterred them from 
examining it As soon as a subject comes to be 
considered 'mathematical' its claims seem gene- 
rally> by the mass of readers, to be either on the 
one hand scouted or at least courteously rejected, 
or on the other to be blindly accepted with all 
their assumed consequences. Of impartial and 
liberal criticism it obtains little or nothing. 

The consequences of this state of things have 
been, I think, disastrous to the students them-» 



PBEFAOE. IX 

selves of Probability. No science can safely be 
abandoned entirely to its own devotees. Its de- 
tails of course can only be studied by those who 
make it their special occupation, but its general 
principles are sure to be cramped if it is not ex- 
posed occasionally to the free criticism of those 
whose main eultiure has been of a more general 
character. Probability has been very much aban- 
doned to mathematicians, who as mathemati- 
cians have generally been unwilling to treat it 
thoroughly. They have worked out its results, 
it is true, with wonderful acuteness, and the 
greatest ingenuity has been shown in solving 
various problems that arose, and deducing subordi- 
nate rules. And this was all that they could in 
fairness be expected to do. Any subject which 
has been discussed by such men as Laplace and 
Poisson, and on which they have exhausted all 
their powers of analysis, could not fail to be pro- 
foundly treated, so far as it fell within their pro- 
vince. But from this province the real principles 
of the science have generally been excluded, or so 
meagrely discussed that they had better have been 
omitted altogether. Treating the subject as ma- 
thematicians such writers have naturally taken it 
up at the point where their mathematics would 
best come into play, and that of course has not 
been at the foundations. In the works of most 



. I 



X PIUBPACS. 

writers upon the subject we should search in vain 
for anything like a critical discussion of the funda- 
mental principles upon which its rules rest, the 
class of inquiries to which it is most properly ap- 
plicable, or the relation it bears to Logic and the 
general rules of inductive evidence. Even in the 
essay of Laplace, a work commonly regarded as 
the principal philosophical text-book on the sub- 
ject, the definition at the outset includes the very 
conception of Probability which it undertakes to 
explain. In the hands of less systematic writers, 
especially amongst those who have treated the 
subject in a popular way, such confusion becomes 
far more serious. One proof only need be given 
to show the utter vagueness and uncertainty with 
which the foundations of the science are frequent- 
ly conceived. Li diflferent books, of which some 
refer to the others as authorities, — ^to a certain 
extent indeed in the same books, — we shall find 
Probability spoken of, sometimes as a property of 
mind, namely, the intensity of the belief with 
which we entertain a proposition; sometimes as 
something external to us which measures this in- 
tensity ; sometimes as an abstract number, namely, 
a numerical fraction. 

This w%nt of precision as to ultimate principles 
is perfectly compatible here, as it is in the de- 
partments of Morals and Politics, with a general 



PB£FACE. XI 

agreement on processes and results. But it is^ to 
say the leasts unphilosophical, and denotes a state 
of things in which positive error is always liable 
to arise whenever the process of controversy forces 
us to appeal to the foundations of the science. I 
shall endeavour to show that this has actually 
been the ca&e, and that confusion and perplexity 
have been thus introduced into some of the most 
important controversies agitated at the present 
day. 

With regard to the remai-ks in the last few 
paragraphs, prominent exceptions must be made 
in the case of two recent works at least* The 
first of these is Professor De Morgan's Formal 
Logic. He has there given an investigation into 
the foundations of Probability as conceived by him, 
and nothing can be more complete and precise 
than his statement of principles, and his deduc- 
tions from them. If I could at all agree with 
these principles there would have been no ne- 
cessity for the following essay, as I could not hope 
to add anything to their foundation, and should be 
far indeed from rivalling his lucid statement of 



* I am here speaking, of course, of those only who have 
expressly treated of the fonndations of the science. Mr Todhunter's 
great work on the History of the Theory of Prdbahility heing, as 
the name denotes, mainly historical, such enquiries have not 
directly fallen within his province. 



XU PBEFACE. 

them. But in his scheme Probability is regarded 
very much from the conceptualist point of view; 
as stated in the preface^ he considers that Proba- 
bility is concerned with formal inferences in which 
the premises are entertained with a conviction 
short of absolute certainty. With this view I 
cannot agree. As I have given a detailed criti- 
cism of some points of his scheme in one of the 
following Chapters, and shall have occasion fre- 
quently to refer to his work, I need say no more 
about it here. The other work to which I refer 
is the profound Laws of Thought of the late Pro- 
fessor Boole, to which somewhat similar remarks 
may in part be applied. Owing however to his 
peculiar treatment of the subject I have scarcely 
anywhere come into contact with any of his ex- 
pressed opinions. 

The view of the province of Probability adopt- 
ed in this Essay differs so radically from that of 
most other writers on the subject, and especially 
from that of those just referred to, that I have 
thought it better, as regards details, to avoid all 
criticism of the opinions of others* except where 
conflict was unavoidable. With regard to that 
radical difference itself Bacon's remark applies, 
behind which I must shelter myself from any 
charge of presumption, — "Quod ad universalem is- 
tam reprehensionem attinet, certissimum vere est 



PBEFACE. Xm 

rem reputanti, earn et magis probabilem esse et 
magis modestam^ quam si facta fuisset ex parte." 

Almost the only writer who seems to me to 
have expressed a just view of the natiire and 
fomidation of the rules of ProbabiUty is Mr Mill, 
in his System of Logic. His treatment of the 
subject is however very brief, and a considerable 
portion of the space which he has devoted to it is 
occupied by the discussion of one or two special 
examples. There are moreover some errors, as it 
seems to me, in what he has written, which will 
be referred to in some of the following chapters. 

The reference to the work just mentioned will 
serve to convey a general idea of the view of 
Probability adopted in this Essay. With what may 
be called the Material view of Logic as opposed to 
the Formal or Conceptualist, — ^with that which 
regards it as taking cognisance of laws of things 
and not of the laws of our own minds in thinking 
about things, — I am in entire accordance. Of 
the province of Logic, regarded from this point 
of view, and under its widest aspect, Probability 
may, in my opinion^ be considered to be a portion. 
The principal objects of this Essay are to ascertain 
how great a portion it comprises, where we are to 
draw the boundary between it and the contiguous 
branches of the general science of evidence, what 
are the ultimate foundations upon which its rules 

b 



3dV ?Bfii*ACK. 

rest, what the nature of the evidence they are ca- 
pable of aflfording, and to what class of subjects they 
inay most fitly be • applied. That the science of 
Probability, on this view of it, contains something 
more important than the results of a system of 
mathematical assumptions, is obvious. I am 
convinced moreover that it can and ought to be 
rendered both interesting and intelligible to or- 
dinary readers who have any taste for philosophy. 
In other words, if the large and growing body of 
readers who can find pleasure in the study of 
books like Mill's Logic and WhewelFs Inductive 
Sciences, turn with aversion from a work on 
Probability, the cause in the latter case must lie 
either in the view of the subject or in the manner 
and style of the book. ^ 

The general design of the following Essay, as 
a special treatise on Probability, is, I think, 
original. Hence, probably, many errors will be 
detected in it, and most certainly many omis- 
sions and imperfections of treatment. Some of 
these might perhaps have been guarded against by 
delaying publication, but however much one might 
be tempted to delay for one's own reputation and 
personal feeling, it is very doubtful whether the 
interests of science are generally best advanced by 
such a course. Provided always that the princi- 
pal ideas and their connection with one another 



PBEFACE< XT 

have been thoroughly thought out, it seems to me 
that one had better bring them out at once for 
others to look at and see what they are worth. 
Hostile criticism is a rough but very efficacious 
way of finding out errors, and a few months of 
such contact with others may in the end do more 
good than would be attained by years of solitary 
reflection. I only hope that those who may 
detect errors and inconsistencies in the following 
pages will not too readily conclude that they are 
a sign of crude thought or over hasty publication. 
No one, until he has actually made the attempt, 
can conceive the prodigious difficulty of thinking 
and writing with perfect consistency upon a sub- 
ject which has been already treated by men his 
superiors in ability and knowledge, but which they 
have discussed from a very different point of view. 
Under such circumstances it is almost vain to 
hope that he can have entirely escaped from what 
he is bound in reason to regard as their mislead- 
ing influence. 

I take this opportunity of thankiDg several 
friends, amongst whom I must especially mention 
Mr Todhimter, of St John's College, and Mr 
H. Sidgwick, of Trinity College, for the trouble 
they have kindly taken in looking over the proof- 
sheets, whilst this work was passing through the 
press. To the former in particular my thanks are 

b2 



XVI PREFACE, 

due for thus adding to the obligations which I, as 
an old pupil, already owed him, by taking an 
amount of trouble, in making suggestions and 
corrections for the benefit of another, which few 
would care to take for anything but a work of their 
own. His extensive knowledge of the subject, and 
his extremely accurate judgment, render the ser- 
vice he has thus afforded me of the greatest pos- 
sible value. 



J. V. 



GONYILLE AND GaIUB CoLLBOE. 

Sqfkmber, 1866. 



TABLE OF CONTENTS. 



GHAPTEB I. 

ON A CBBTAIN KIND OF SSBIES AS THB FOUNDATION OF 
PEOBABILITY. 

§§ I — 3^ The foundatioii of Probability, on which all its rulee are 
based, ifl a certain kind of series, which combines 
aggregate regularity with individiud iiregularity. 

4. The objects composing the series are irregular in certain 

respects only, 

5. And may occur in any order in time. 

6. 7. Substitute for the phrases 'an event* and ' the ways in 

which it can happen,' the conception of a series 
whose terms are combined by peQQttQgntuttributeSy 
and distinguished, ih definite proportions, by varia- 
ble attributes. 

8. Wetlius include the common phrases. 

9. No occasion to analyse beyond this series. 

10. II. The aggregate regularity^ when ^camined on a very 

great scale, is generally found to be itself irregular ; 
ie. the 'type ' or 'mean' is not fixed. 

1 3. Though in games of chance it does seem fixed. 
. 13* Hence two kinds of series. 

14. ffince the fixed series only is fit for accurate rules of 

■dence, we often have, in calculation, to substitute 
one of this kind. 

15. Physical illustration to explain the way in which' 

mathematics are used in Probability. 

16. (i) Experience only can shew over what extent the 

mathematics, are applieaUe^ <a) The mathematics^ 



XVIU CONTENTS, 

themselyes, however, have no limit. (3) No sudden 
break between the objects to which the mathematics 
do and do not apply. 
17. The two kiiids of series illustrated. 

CHAPTER II. 

THE 8EEIBS OP PROBABILITY ARE OBTAINED BY EXPERIENCE 
AND NOT X PRIORI, AND THEY OCCUR IN GROUPS. 

§§ I, 2. The experience by which the series is obtained is ex- 
tended, of course, by Induction. 

g. A common theoiy is that the series can be obtained 

dprion. 

4, 5. Examination of this theory in the ease of tossing up a 
penny, and proof that the experimental fact of a 
series is tacitly involved even here. 

6. Narrow applicability of the d priori theory, 

7. Though frequent advantage of d priori calculatioDS. 

8. g* Phase of this theory adopted by Laplaoe, 
f a And in the Theorem of BemouillL 

11. This theory is utterly inapplicable to most social 

subjects, 

1 2. And leads to considerable ezror. 

13. The regularity of averages cannot be demonstrated by 

arithmetic. 

14 — 16. Quotation from Quetelet on the difference between 
meant and averages, with comments thereon. 

1 7 — 19. The important distinction is between a type (or mean), 
and a group of types. The latter is what we afa^ioet 
always find in nature, though there seems no reason 
for believing that all such groups must be identical. 

10. Is such identity proved by the Bule of L^ast Squares? 

21. ninstration of some of the preceding remarks by 

means of the Petersburg Problem, 

94. Summary of results obtained. 



CONTENTS* xix 

CHAPTER III. 
QRABATIONS OF BELIEF. 

1 1. Description of the position occupied at tbis point, 

2. We have now to inquire whether inferences can be 

drawn about the detaUa of the serieB ah^ady de- 
scribed. 

3, 4. It 18 a common doctrine that the subject-matter of 

Probability is ' quantity of belief.' 

5. This doctrine, eren if correct, would be inadequate. 

6. But is it correct ?— Two objections stated. 

7. (i) Difficulty of measuring the amount of our belief 

owing to the disturbing influence of emotions, 

8. And owing to the complexity of the evidence for 

every proposition. 

9. {2) Our belief, so far as it can be measured, is not 

naturally what theory would assign. 
ID. If we have instincts of belief, must they be correct f 

II. Objection — * If our belief is not of a certain amount, 

it ought to be.' Yes ; It will often need correction 

by experience. 
13. The Analogy of Formal Logic does not support the 

doctrine in question. 

13. Summary of preceding sections. 

14. But does not every one recognize the fact of grada- 

tions of belief ? 
15 — 20, Detailed examination into the interpretation of this 
. partial belief when we are concerned with material 

logic, or inference about things. 
21 — 23. This interpretation capable of wider application than 

appears at first sight. 
14. Summary of results obtained. 

35* Limits within wl^ich causation is demanded. 



XX CSONTBNTS. 

27. Confasion caused by the attempt to justify not only 

the belief, but the emotions which accompany it. 

28. Illustration of this in the Petersburg Problem. 

29. 30. Our surprUe, however, does admit of some justification. 

31. Illustration^ in common language, of the two sides of 

Probability, the objective and the subjective. 

32. Advantage of discussing the subjective side. 

33. Definition of the phrase Hhe chance of an event.^ 

CHAPTER IV. 
THE EULES OP INFERENCE IN PEOBABILITY. 

§§ I, 2. The inferences considered in the last chapter corre- 
spond to immediate inferences in Logic. We now 
pass on to syllogistic inferences. 

3. (i) Inferences made by addition and subtraction. 

4. (2) Inferences made by multiplication and division* 

5. 6. (3) Ezamination of the common rule for inferring the 

probability of the combination of two independent 

events. 
7. Contrast of the above view of the rules of inference 

with the view of Professor De Morgan. 
8 — 10. Other rules, besides the above, are often introduced. 

CHAPTER V. 

GENERAL REMARKS ON THE RESULTS OF THE FOREGOING 
CHAPTERS. 

§§ I — ^4, Brief description of the standing point occupied on 
the material view of Probability, and illustration 
of it from the corresponding view of Logic. 

5. We are not concerned with time in Probability. 

6 — II. Examination of the doctrine that there is a differ- 
ence between probability before and after the iiact. 

12. Origin of this doctrine. 



CONTENTS. XXI 

13 — 15. Examples in confirmation of some of the above state- 
ments, and common explanation of these examples. 
16, 17. Doctrine of the reiatiyity of probability. 



CHAPTER VI. 
THE BULE OP SUCCESSION. 

§ I, 2. Probability and Induction closely connected. 
3, 4. Origin of the common confusion between them. 

5. In judging of the probability of any particular event 

two distinct causes influence our conviction : 

6, 7. These belong respectively to Probability and Induc- 

tion. 
8, 9. Reasons for confining the province of Ptobability. 

10. Statement of the Rule of Succession. 

11. Its general acceptance. 
11, Outline of criticism of it. 

13. This Rule not the expression of a mere instinct. 

14. Examples to test the correctness of the rule. 

{5. The 'rule not to be defended on the plea that it is 

for those who have no other knowledge. 
16. Professor De Morgan's defence of the rule. 

17 — 19. Other defences. 

20. How such a rule is really to be regarded. 

21, 22. Whence is the ordinary rule obtained? 
23. More complicated forms of the rule. 

CHAPTER VII. 

INDUCTION, AND ITS CONNECTION WITH PROBABILITY. 

§ I. . Induction necessary for any inference about things. 
2 — 5. Brief examination of the nature of Inductive infer- 
ence. 
6. Boundary between Induction and Probability. 



XXll CONTENTS* 

7. Beasons of the limits within which oausation is 

needed in ProbahiUty. 

8, 9. Induction having given us generalizations, how is 

Probability to use them ? 

fo, II. Wherein the step before us difiEers from the corre- 
sponding step in ordinary material logic. 

12 — 14. Two-fold perplexity in the process of inference arising 
from things being comprised in many different 
classes, (i) Mild form of this perplexity. 

15. (1) Aggravated form. 

f 6. No theoretical difficulty in such perplexity. 

17. Blnstration from Life Insurance. 

18. The practical difficulty is of a quite distinct kind. 

19. 90. Consistency of the above-mentioned perplexity with 

our view of Probability. 

21. Summary of results obtained. 

22. Remarks on making inductive anticipations. 

23. Mr Mill's view of Induction. 

24. Dr Whewell's view ; forming ' conceptions. ' 

25. The difficulty of forming the conceptions, 

26. And their indeterminateness. 

27. Bearing of foregoing considerationB upon rules of 

anticipation. 



CHAPTEB VIIL 
ON DIRECT AND INVERSE PROBABILITY. 

; I . Statement of the distinction in question. 

2, 3. Examples shewing its unimportance. 

4) 5. Arbitrary element in many artificial problems. 

6. Contrast with these an example from nature. 

7. Distinction between appropriate and inappropriate 

applications of Probability. 



CONTENTS. ZXUV 

CHAPTER IX, 

CRITICIBM or SOME OOlffMON CONCLUSIONS IN PBOBABILITY. 

§ I. Examples illustrative of the question at iMue. 

9, 3. The applicability, not the acciympy, of the mathe- 
matics doubted. 

4. Inapplicability of the Rale of Sufi^cieQt Reaaon for 

the purpose to which it is often applied. 

5, 6. Can experience be appealed to for this purpose t 

7. ProTince of mathematics in Probability. 

8. Example from Quetelet, 

CHAPTER X. 
THE APPLICATIOJT OF PEOBA^ILITY TO TESTIMONY. 

§ I. Doubtful applicability of Probability to testimony. 

2. Whence the objections to such application arise. 

3, 4. Conditions under which these objections would fail. 

5. Reasons for the above conditions. 

6, 7 . Are these conditions fulfilled in the case of testimoDy ? 

8. The appeal here is not really to statistics. 

9. Additional illustration. 

10. Objection as stated by Mr Mill. 

If. Is any application of Probability to testimony valid ? 

CHAPTER XI. 

ON THE CAUSES BT WHICH THE PECULIAB SEMES OF 
PB03ABILITY AI^ PBODUCED. 

§§ I, 2. Division of these causes into objects and agencies. 

3. Property observable in these causes, 

4. Similar to that observable in the series themselves, 

5. (i) Uniformity existing amongst the objects. 

6. (3) Uniformity in the agencies affecting the objectB. 



Xxiv CONTENTS. 

7. These unifoimities purely experimental, and not 

fixed, nor found univerBally. 

8. In what ehuises ci things do these unifonnities exist? 

9. Amongst natural objects only. 

10, II. Apparent exceptions to this (i) Bub of Least 
Squares. 

12. (2) Games of chance. 

.13 — 15. The above uniformities not found in artificial as op- 
posed to natural results. 



CHAPTER XII. 
FALLACIES. 

§ I. Advantage of classifying fallades. 

2. (I.) Judgments formed after the event. 

3, 4. Example to shew the different problems often con- 

fused in such judgments ; — 

5. (1) The probability of the event, 

6. (2) That of the witness speaking truth, 

7. (3) That of the event arising in a certain way. 

8. 9. Illustrations of the above distinctions. 

10. (II.) Undue limitations of the notion of probability. 

11, 12. Illustrations from narrow escapes. 

13. (III.) Confusion between rare and impossible events. 

14. Origin of this confusion. 

15. 16. Production of Shakespeare by chance. 

17. Explanation of the above. 

18. (IV.) Confusion between Probability and Induction. 

19. (i) Past recurrence may increase our expectation, 

20. (2) Or leave it unaffected. 

2 1 . Explanation of the above. 

22. (3) Or diminish expectation. 

23. Example in illustration. 



OGHfTESTS, XXV 

CHAPTER XIII. 
ON THE CREDIBILITY OF EXTRAOBDINABT STOBIE& 

§ I. Introduction to the enquiry. 

1. Two conditions under which the credibility of a story 

is independent of the nature of the story. 

3. (i) Statement of one of these conditions. 

4. ' Contest of opposite improbabilities.* 

5. The contest may be evaded ; 

6. Cr it may be encountered. 

7. The result indeterminate in the latter case, 

8. Though various solutions are offered. 

9 — II. Explanation and illustrations of the abov^-mentioned 

indeterminateness. 

13. (4) Statement of the second oondition. 

13, 14. Example in illustration. 

15. Meaning of the term ' improbable.' 

16. Simimary of results. 

1 7. Combination of testimony. 

18. Probability not strictly concerned with miraculous 

accounts. 

ig, 20. Description of a miracle. 

9 1. Two distinct prepossessions in regard to miracles. 

22. Meaning of prepossessions. 

^3. Consequences of their existence. 

24, Example in illustration. 

25, Inadequate recognition of these considerations. 

26, Hence the futility of many arguments. 
37. Summary of results. 

CHAPTER XIV. 
CAUSATION. 

§ I. Disputes arising out of the doctrine of causation. 

2. Definition of the term cause. 

3. Two elements in this definition. 



XXVI contents: 

4, 5. First departure from the definition. 

6. Second departure. Distinction between empirical 

uniformities and laws of causation. 

7. This distinction theoretically unsound. 

8. Origin of the distinction. 

9. Examples in illustration. 

10, II. Meaning of tf7ime<2ta^ sequence. 

12. Bearing of the above resultCi on the doctrine of caus- 

ation. 

1 3. The term causation ambiguous. 

14. Is causation demanded in Probability t 

15. Can causation be proved by statistics ? 

16. Singularity of the examples sometimes adduced. 

17. Irrelevance of the above enquiries to Probability. 

18. Real nature of the question at issue. 

19 — 22. Statistics, as commonly given, do not prove causation. 

23. What such statistics do prove. 

24. Freedom of the WilL 

25. Theological inferences traba. Probability; — (i) that 

events happen by chance. 

26. (2) That events are only the development of their 

respective probabilities. 

27. Different feelings that might be excited by observa- 

tion of the same statistics. 



CHAPTER XV. 
ON STATISTICS AS APPLIED TO HUMAN ACTIONS. 

§ I. Are statistics as applicable to the actioiw d m^n as 

to inanimate things 1 

2. Statement of two conditions for cuch applicability. 

3. (t) The objects in the statistics must be left undisturbed. 

Does this hold in the case of human actions ? 

4. Quotation from Mill's Logic, 



1 



OONTENTS. XXVU 

5, 6. The statistics may be disturbed by publishing our 

calculations. 
7, 8. Is this disturbance practically intrignificant t 
9. (2) The observer must not intrude himself amongst 

the statistics. 

10. Characteristics of these statistioi* 

11. Why the observer is liable to intrude himself into 

the statistics. 

12. Insignificance of such actual intrusion. 

13. But confusion caused by hypothetic intrusions. 

14. Quotation from Buckle's Hutory of Civilization. 
\'$: Prevalence of the opinions in this quotation. 
16,17. Kemarks upon this quotation. 

18 — 20. Confusion in it between (i) the speculative, and (2) 
the practical standing-point. » 

21. Whence the common errors about the power of 

social laws, 

2 2 . And the complaints about the impotence of individual 

efforts. What these complaints really mean. 

23. Conclusion. 



ERRATA. 

Page 35, for Bernoulli read BernouIIli 
„ 91, line four from bottom, fw twenty-five nod thirty-four 
„ 119, first line, for or read and 



THE LOGIC OF CHANCE. 



CHAPTER I. 

ON A CERTAIN KIND OF SERIES AS THE 
FOUNDATION OF PROBABILITY. 

§ 1. It is not generally easy to give a clear defini- 
tion of a science at the outset, so as to set its scope 
and province before the reader in a few words. In the 
case of those sciences which are more immediately and 
directly concerned with objects, this difficulty is not 
indeed so serious. If the reader is already familiar 
with the objects, a simple reference to them will give 
him a tolerably accurate idea of the direction and 
nature of his studies. Even if he be not familiar 
with them, they will still be often connected and 
associated in his mind by a name, and the mere state- 
ment of the name will convey a fair amount of preli- 
minary information. This is more or less the case 
with many of the natural sciences ; we can often tell 
the reader beforehand exactly what he is going to 
study. But when a science is concerned, not so much 
with objects directly, as with processes and laws, or 
when it takes for the subject of its enquiry some 
conaparatively obscure feature drawn from phenomena 

1 



2 THE LOGIC OP CHANCE. [CHAP. I, 

whicli have little or nothing else in common, the 
difficulty of giving preliminary information becomes 
greater. Recognised classes of objects have then to 
be disregarded and even broken np, and an entirely 
novel arrangement of the objects to be made. In such 
cases it is the study of the science that first gives the 
science unity, for till it is studied the objects with 
which it is concerned were probably never thought 
of together. Here a definition cannot be given at the 
outset, and the process of obtaining it may become 
somewhat laborious. 

The science of Probability, at least on the view 
taken of it in the following pages, is of this latter 
description. The reader who is at present unac- 
quainted with it cannot be at once informed of its 
scope by a reference to objects with which he is al- 
ready familiar. He must be taken in hand, as it 
were, and shown the objects before he will know 
them. To do this will be our first task. 

§ 2. In studying Nature, in any form, we are con- 
tinually coming into possession of information which 
we sum up in general propositions. Now in very 
many cases these general propositions are neither 
more nor less certain and accurate than the details 
which they embrace and of which they are composed. 
I am assuming at present that the truth of our gene- 
ralizations is not disputed ; as a matter of fact these 
generalizations may rest on weak evidence, or they 



1 



SBcrr. 2.] the logic op chance. 3 

may be uiicei*taiii from their being widely extended 
by Induction ; what I mean is, that when we resolve 
them into their component parts we have precisely 
the same assurance of the truth of the details as we 
have of that of the whole. When I know, for in- 
stance, that cows ruminate, I feel just as certain that 
any particular cow or cows ruminate as that the 
whole class does, I may be right or wrong in my 
original statement, and I may have obtained it by 
any conceivable mode; but whatever the value of the 
general proposition may be, that of the particulars is 
neither greater nor less. If one of these ' immediate 
inferences' is justified at all, it will be always right. 

But it is by no means necessary that this should 
be the case. There is a class of immediate inferences, 
unrecognized indeed in Logic, but constantly drawn 
in practice, of which the characteristic is, that as they 
increase in particularity they diminish in certainty. 
Let me assume that I am told that some cows rumi- 
nate j I cannot infer logically from this that any par- 
ticular cow does so, though I should feel some way 
removed from absolute disbelief or even indifference 
on the subject; but if I saw a herd of cows I should 
feel more sure that some of them were ruminant than 
I did of the single cow, and my assurance would in- 
crease with the numbers of the herd about which I 
had to form an opinion. Here then we have a class 
of objects as to the individuals of which we feel quite 

1—2 



4 THE LOGIC OF CHANCE. [CHAP. I. 

in uncertainty, whilst as we embrace larger numbers 
in our assertions we attach greater weight to our in- 
ferences. It is with JBuch classes of objects and such 
inferences that the science of Probability is concerned. 

§ 3. In the foregoing remarks, which are intended 
to be purely preliminary, I have not been able alto- 
gether to avoid some reference to a subjective element, 
viz. the degree of our certainty or belief about the 
things which we are supposed to contemplate. The 
reader may be aware that by some writers this element 
is regarded as the subject-matter of the science. Hence 
it will have to be discussed in a future chapter. As 
however I do not at all agree with the opinion of the 
writers just mentioned, I shall make no further allu- 
sion to it here, but pass on at once to a more minute 
investigation of that distinctive characteristic of cer- 
tain classes of things which was introduced in the 
last section. 

In these classes of things, which are those with 
which Probability is concerned, the fundamental con- 
ception which the reader has to fix in his mind as 
clearly as possible, is, I take it, that of a series. But 
it is a series of a peculiar kind, one of which I can 
see no better compendious description than that which 
is given by the statement that it combines individual 
irregularity with aggregate regularity. This is a 
statement which will probably need some explanation. 
Let us recur to an example of the kind already 



^mBmemmmm^mmmmgmmm'^^^^^^^f^ - J i » ^y-.,^ sm^^^^ 



.SECT. 3.] THE LOGIC OF CHANGE. 5 

alluded to, selecting one which shall be in accordance 
with experience. Some cows will not suckle their 
young. Now if this proposition is to be regarded as 
a purely indefinite or, as it would be termed in logic, 
particular proposition, no doubt the notion of a series 
does not obviously present itself in connection with 
it. It contains a statement about a certain unknown 
proportion of the whole, and that is all. But it is 
not with these purely indefinite propositions that we 
shall be concerned. Let us suppose the statement, 
on the contrary, to be of a numerical character, and 
to refer to a given proportion of the whole, and we 
shall then find it difficult to exclude the notion of a 
series. We shall find it, I think, impossible to do so 
as soon as we set before us the aim of obtaining accu- 
rate or even moderately correct inferences. What, 
for instance, is the meaning of the statement that 
one cow in ten fails to suckle its young? It certainly 
does not declare that in any given herd of, say twenty, 
we shall find just two that fail : whatever might be 
the strict meaning of the words, this is not the im* 
port of the statement. It rather contemplates our 
examination of a large number, of a long succession 
of instances, and states that in such a succession we 
shall find a numerical proportion, not indeed accurate 
at first, but which tends in the long run to become 
accurate. In every kind of example with which we 
shall be concerned we shall find, I think, this referr 



6 THE LOGIC OP OHANCK. [COAV. I. 

ence to a large number or succession of objects, or, as 
I shall term it, series of them. 

A few additional examples may serve to make 
this plain. 

Let us suppose that we toss up a penny a great 
many times ; the results of the successive throws may 
be conceived to form a series. The separate throws 
of this series seem to occur in utter disorder; it is 
this disorder which causes our uncertainty about 
them. Sometimes head comes, sometimes tail comes; 
sometimes there is a repetition of the same face, 
sometimes not. So long as we confine our observa- 
tion to a few throws at a time, the series seems to be 
«imply chaotic. But when we consider the result of 
a loDg succession we find a marked distinction; a 
kind of order begins gi*adually to emerge, and at last 
assumes a distinct and striking aspect. We find in 
this case that the heads and tails occur in about equal 
numbers, that similar repetitions of different faces do 
fio also, &c. In a word, notwithstanding the indi- 
vidual disorder, an aggregate order begins to prevail 
So again if we are examining the length of human 
life, the different lives whicli fall under our notice 
compose a series presenting the same featui^es. The 
length of a single life is proverbially uncertain, but 
the average duration of a batch of lives is becoming 
in an almost equal degree proverbially certain. The 
larger the number we take out of any mixed crowds 



• SECT. 4.] THE LOGIC OP CHANCE. 7 

the clearer become the symptoms of order, the more 
nearly will the average length of each selected class 
be the same. These few cases will serve as simple 
examples of a property of things which can be traced 
almost everywhere, to a greater or less extent, through- 
out the whole field of our expeiience. Fires, ship- 
wrecks, yields of harvest, births, marriages, suicides; 
it scarcely seems to matter what feature we single 
out for observation. The irregularity of the single 
instances diminishes when we take a large number, 
and at last seems for all practical purposes to disap- 
pear. 

§ 4. In speaking as above of events or things as 
to the details of which we know little or nothing, it is 
not of course implied that our ignorance about them 
is complete and universal, or, what comes to the 
same thing, that irregularity may be observed in all 
their qualities. All that is meant is that there are 
Home qualities or marks in them, the existence of 
which we are not able to predicate in the individuals. 
With i-egard to all their other qualities there may be 
the utmost uniformity, and consequently the most 
complete certainty. The irregularity in the length of 
human life is notorious, but no one doubts the exist- 
ence of a heart and brains in any person whom he 
happens to meet. And even in the qualities in which 
the irregularity is observed, there are often, indeed 
generally, positive limits within which it will be 



8' THE LOGIC OP CHANCE. [CHAP. I. 

found to be confined. No person, for instance, can 
calculate what may be the length of any particular 
life, but we feel perfectly certain that it will not 
stretch out to 150 years. The irregularity of the in- 
dividual instances is only shewn in certain respects, 
as a g. the length of the life, and even in these it has 
its limits. The same remark will apply to most of 
the other examples with which we shall be concerned. 
The disorder in fact is not universal and infinite, it 
only prevails in certain directions and up to a certain 
point. 

§ 5, In speaking as above of a series, it will 
hardly be necessary to point out that we do not imply 
that the objects themselves which compose the series 
must occur successively in time; the series may be 
formed simply by their coming in succession under our 
notice, which as a matter of fact they may do in any 
order whatever. A register of mortality, for instance, 
may be made up of deaths which took place simulta- 
neously or successively; or we might if we pleased 
arrange the deaths in an order quite distinct from 
either of these. This is entirely a matter of indifier- 
ence; in all these cases the series, for any purposes 
which we need take into account, may be regarded as 
being of precisely the same description. The objects, 
be it remembered, are given to us in nature; the 
order under which we view them is our own private 
arrangement, I mention this here simply by way of 



SECT. 6.] THB LOGIC OP CHANCE. 9 

caution, the meaning of this assertion will become 
more plain in the sequel. 

§ 6. The reader will now have in his mind the 
conception of a series of things or events, of the in- 
dividuals of which we know but little, whilst we £nd 
a continually increasing uniformity as we take larger 
numbers under our notice. This is definite enough 
to point out tolerably clearly the kind of things with 
which we have to deal, but it is not sufficiently definite 
for purposes of accurate thought. We must therefore 
attempt a somewhat closer analysis. 

There are certain phrases so commonly adopted, 
as to have become part of the technical vocabulary 
of the subject, such as an ' event' and the ^ way in 
which it can happen.' Thus the act of throwing a 
penny would be called an event, and the fact of its 
giving head or tail would be called the way in which 
the event happened. If we were discussing tables of 
mortality, the former term would denote the mere 
fact of death, the latter the age at which it occurred, 
or the way in which it was brought about, or what- 
ever else in it might be the particular circumstance 
under discussion. This phraseology is very conve- 
nient, and I shall often make use of it, but without 
explanation it may lead to confusion. For in many 
cases the way in which the event happens is of such 
great relative importance, that according as it happens 
in one way or another the event would have a different 



10 THE LOGIC OP CHANCIE. [CHAP. I. 

name, in other words, would or would not be the same 
event. The phrase therefore will have to be consider- 
ably stretched before it will conveniently cover all the 
cases to which we may have to apply it If we were 
contemplating a series of human beings, male and 
female, it would sound odd to call their humanity an 
event, and their sex the way in which the event hap- 
pened. If we recur however to any of the classes of 
objects already referred to, we may see our path to- 
wards obtaining a more accurate conception of what 
we want. It will easily be seen that in every one of 
them there is a mixture of similarity and dissimilarity; 
there is a series of events which have a certain number 
of features or attributes in common, — without this 
they would not be classed together. But there is also 
a distinction existing amongst them ; a certain number 
of other attributes are to be found in some and are not 
to be found in others. In other words, the individuals 
which form the series are compound, each being made 
up of a collection of things or attributes; some of 
these things exist in all the members of the series, 
others are found in some only. So far there is nothing 
peculiar to the science of Probability ; that in which the 
distinctive characteristic consists is this; — that the 
occasional attributes, as distinguished from the perma- 
nent, are found on an extended examination to exist 
in a eertmn definite proportion of the whole number of 
cctaes. We cannot tell in any given instance whether 



SECT. 7.] THE LOGIC OP CHANCE. 11 

they will be found or not, but as we go on examining 
more cases we find a growing uniformity. We find 
that the proportion of instances in which they are 
found to instances in which they are wanting, is gra- 
dually subject to less and less variation, and ap- 
proaches continually towards some apparently fixed 
value. 

The above is the most comprehensive foim of de- 
scription ; as a matter of fact the groups will in many 
cases take a far simpler form, they may appear, e. g. 
simply as a succession of substances of the same kind, 
say cows, with or without an occasional attribute, 
say redness. I am using the word attribute, of course, 
in its widest sense, intending it to include every dis- 
tinctive feature that can be observed in a thing, from 
essential qualities down to the merest accidents of 
time and place. 

§ 7. On examining our series, therefore, we shall 
find that it may best be conceived, not as a succes- 
sion of events happening in different ways, but as a 
succession of groups. These grouj^s, on being analysed, 
are found in every case to be resolvable into collections 
of substances and attributes. That which gives its 
unity to the succession of groups is the fact of some of 
these substances or attributes being common to the 
whole succession; that which gives their distinction to 
the groups in the succession is the &ct of some of 
them containing only a portion of these substances and 



12 THE LOGIC OF CHANCE. [CHAP. I. 

attributes, the other portion or portions being occasion- 
ally absent. So understood, I think our phraseology 
will embrace every class of subjects of which Proba- 
bility can take account. 

§ 8. It will be easily seen that the ordinaiy ex- 
pression is included in the one adopted above. When 
the occasional attributes are unimportant the per- 
manent ones are sufficient to fix and appropriate the 
name, the presence or absence of the others being simply 
denoted by some modification of the name or the 
addition of some predicate. We may therefore in all 
such cases speak of the collection of attributes as ' the 
event,* — the same event essentially, that is — only say- 
ing that it (so as to preserve its nominal identity) 
happens in different ways in the different cases. When 
the. occasional attributes however are important^ or 
compose the majority, this way of speaking becomes 
less appropriate; language is somewhat strained by 
our implying that two extremely different assemblages 
are in reality the same event, with a difference only in 
its mode of happening. The phi'ase is however a very 
convenient one, and with this caution against its being 
misunderstood, I shall frequently make use of it. 

§ 9. A series of the above-mentioned kind is, I 
apprehend, the ultimate basis upon which all the rules 
of Probability must be based. It is essential to a clear 
comprehension of the subject to have carried our 
analysis up to this point, but any attempt at further 



SECT. 10.] ^ THE LOGIC OP CHANCB. 13 

analysis into the intimate nature of the events com- 
posing the series, is not required. It is not necessary, 
for instance, to form any opinion upon the questions 
discussed in metaphysics as to the independent ex- 
istence of substances. We have discovered, on exa- 
mination, a series composed of groups of substances 
and attributes, or of atttibutes alone. At such a series 
we stop, and thence investigate our rules of evidence; 
into what these substances or attributes would them- 
selves be ultimately analysed it is no business of ours 
to enquire here. 

§ 10. The stage then which we have now reached 
is that of having discovered a quantity of things (they 
prove to be groups on analysis) which are capable of 
being classified together, and are best regarded as a 
series. The distinctive peculiarity of this series is our 
finding in it an order, gradually emerging out of dis- 
order, and showing in time a marked and unmistake- 
able uniformity. The impression which may possibly 
be derived from the description of such a series, and 
which the reader will probably already entertain if 
he have studied Probability before, is that the gra- 
dual evolution of this order is indefinite, and its ap- 
Mroach therefore to perfection unlimited. And many 
of the examples commonly selected certainly tend to 
confirm such an impression. But in reference to the 
theory of the subject it is, I am convinced, an error, 
and one fraught with confusion- 



H THE LOGIC OF CHANGS. [CHAP. I. 

The lines which have been prefixed as a motto to 
this work, " So careful of the type she seems, so care- 
less of the single life," are soon after corrected by the 
assertion that the type itself, if we regard it for a 
long time, changes and then vanishes and is succeeded 
by othera. So in Probability; that uniformity which 
is found in the long run, offering so great a contrast 
to the individual disorder, though durable is not ever- 
lasting. Keep on watching it long enough, and it 
will be found almost invariably to fluctuate, and in 
time may prove as utterly irreducible to rule, and 
therefore as incapable of prediction, as the individual 
cases themselves. The full bearing of this fact upon 
the theory of the subject, and upon certain common 
modes of calculation connected with it, will appear 
more fully in some of the following chapters; at pre- 
sent I shall confine myself to establishing and illus- 
trating it. 

Let us take, for example, the average duration of 
life. This, provided our data are sufficiently exten- 
sive, is known to be tolerably regular and uniform. 
This has been fully illustrated in the preceding sec- 
tions, and is a truth indeed of which the popular 
mind has a tolerably olear grasp at the present day. ^ « 
But a very little consideration will show that there 
may be a superior as well as an inferior limit to the 
extent within which this uniformity can be observed. 
At the present time the average duration of life in 



SPCT. 11.] THE LOGIC OF CHANCE. 15 

England may be, say thirty; but a century ago it 
was decidedly leas; several centuries ago it was very 
nmcb less; whilst if we possessed statistics referring 
to our early British ancestors we should probably find 
that there has been since that time a still more marked 
improvement. What may be the future tendency no 
man can say for certain. It may be, and we hope 
will be the case, that owing to sanitary and other im- 
provements, the duration of life will go on increasing 
steadily; it is quite conceivable that it should do so 
without limit. On the other hand, this duration 
might gradually tend towards some fixed length. Or, 
again, it is perfectly possible that future generations 
might prefer a short and a merry life, and therefore 
reduce their average. All that I am concerned to 
indicate is, that this uniformity (as we have hitherto 
called it) has varied, and, under the influence of future 
eddies in opinion and practice, may vary still; and 
this to any extent, and with any degree of irregu- 
larity. To borrow a term from Astronomy, we find 
our uniformity subject to what might be called an irre- 
gular secula/r variation. 

§ 11. The above is a fair typical instance. K we 
had taken a less simple feature than the length of life, 
or one less closely Connected with what may be called 
the great permanent uniformities of nature, we should 
have found the peculiarity under notice exhibited in 
a far more sticking degree. The deaths from small- 



16 THE LOGIC OP CHANCE. [CHAP. I. 

pox, for example, or the instances of duelling or ac- 
cusations of witchcraft, if examined during a few 
successive years, would have shown a very tolerable 
degree of uniformity. But this uniformity has risen 
probably from zero; after various and very great fluctu- 
ations seems tending towards zero again; and may, for 
anything we know, undergo still greater fluctuations 
in future. Now these examples I consider to be only 
extreme ones, and not such very extreme ones, of 
what is the almost universal rule in nature. I shall 
endeavour to show that even the few apparent ex- 
ceptions, such as the proi>ortions between male and 
female births, &c., may not be, and probably in 
reality are not, exceptions. A type that is persistent 
and invariable is scarcely to be found in nature. The 
full import of this conclusion will be seen in a future 
chapter. I would only call attention here to the 
important inference that, although statistics are no- 
toriously of no value unless they are in sufficient 
numbers, yet it does not follow but that we may have 
too many of them. If they are made too extensive, 
they may again fall short, at least for any particular 
time or place, of their greatest attainable accuracy. 

§ 12. These natural uniformities then are found 
at length to be subject to fluctuation. Now contrast 
with them any of the uniformities afibrded by games 
of chance; these latter seem to show no trace of 
secular fluctuation, however long we may continue - 



SECT. 14.] LOGIC OF CHANCE. 17 

our examination of them. Some criticism will be 
offered, in the coarse of the next chapter, upon some 
of the common attempts to prove it priori that there 
must be this fixity in the uniformity, but of its exist- 
ence thei-e can scarcely be much doubt. Pence give 
heads and tails alternately now, as they did when they 
were first tossed, and as we believe they will continue 
to do so long as the present order of things continues. 
The fixity of these uniformities may not be as absolute 
Bs is commonly supposed, but no amount of expe- 
rience which we need take into account is likely to 
shake them. Whereas natural uniformities at length 
fluctuate, those afforded by games of chance seem fixed 
for ever. 

§ 13. Here then are series apparently of two differ- 
ent kinds. They are alike in their initial irregularity, 
alike in their subsequent regularity; it is in what we 
may term their ultimate form that they begin to di- 
verge from one another. The one tends without any 
permanent variation towards a fixed numerical propor- 
tion in its uniformity; in the other the uniformity 
is found at last to fluctuate, and to fluctuate, it may 
be, in a manner utterly irreducible to rule. 

§ 14. As this chapter is intended to be simply 
explanatory and illustrative of the foundations of the 
science, I may remark here (what will receive its sub- 
sequent justification) that it is in the case of series of 
the former kind only that we are able to make any- 

■ 2 



18 LOGIC OF CHA17CE. [CHAP. I. 

thing wliicli can be interpreted into strict scientific 
inferences. We shall be able however to see the 
kind and extent of error that would be committed if 
in any example we were to substitute an imaginary 
series of the former kind for any actual series of the 
latter kind which experience may present to us. The 
two series are of course to be alike in all respects, ex- 
cept that the varial?le uniformity has been replaced by 
a fixed one. The difference then between them would 
not appear in the initial stage, for in that stage the 
distinctive characteristics of the series of Probability 
are not apparent; nor would it appear in the sub- 
sequent stage, for the real variability of the uniform- 
ity has not for some time scope to make itself per- 
ceived. It would only be in what we have called 
the ultimate stage, when we suppose the series to ex- 
tend for a very long time, that the difference would 
begin to make itself felt. The numbers of persons, for 
example, who die each year at the age of six months 
are, when, examined on a small scale, utterly irre- 
gular; they become however regular when examined 
on a larger scale ; but if we continued our observation 
for a very great length of time, or over a very great 
extent of country, we should find this regularity itself 
changing in an irregular way. The substitution just 
mentioned is equivalent to saying. Let us assume that 
the regularity is fixed and permanent. It will appear 
in a future chapter that such a substitution has to be 



•^ 



SECT. 15.] LOGIC OF CHANCE. 19 

made in very many examples before thej can become 
fit subjects for strict scientific inference. 

§ 15. As the theory of Probability is almost under- 
stood when the foregoing conceptions are fully grasped, I 
will add, for additional clearness, a physical illustration. 
It will be worth while to work this out rather fully, 
so as to make it serve once for all as a sort of stand- 
ard illustration to refer back to. I might indeed call 
it an analogy rather than an illustration, as it is in 
reality an application of the same principle to geo- 
metry instead of to arithmetic. Let the reader then 
picture to himself a mountainous country, the surface 
of which is all broken up into crag and boulder and 
gully. If we wishM to measure the surfsLce of this 
country we should find mathematics of very little use 
directly; we could only give rough guesses or succeed 
at last by the tedious process of measuring almost 
every patch of a few square yards separately. But in 
parts of it — at the bottom of long broad valleys for 
example — we should find the features of the country 
altogether altered . Here too (which it is very import- 
ant to mark) there would be roughness and irregu- 
larity of surface when we confined our view to a 
small spot ; to a short-sighted man it would still ap- 
pear as if he were standing in the midst of a chaos ; 
but when we took a broader view we should discover 
a regularity that in contrast with the broken hill-side 
would seem almost startling. The greater the extent 

2—2 



20 LOGIC OP CHANCE. [cHAP. T. 

of country over which we cast our eye the more per- 
fect would appear this regularity, till in the far dis- 
tance the soil seemed smoothed down as though with a 
spirit-level. Such a characteristic as this at once en- 
ables us to make use of mathematics : if the bottom 
of the valley be a plane it mu»t possess the properties 
of a plane, and hence the work of mensuration would 
become easy. It is not that we have made any arbi- 
trary assumptions; in examining the face of nature 
we have found that a striking difference does in fact 
exist between one part and the other, by which the 
principles of geometry are enabled to serve our pur- 
pose by a simple and immediate application, lu 
other words, although there is disorder over any very 
small space, and disorder again of another kind over 
a very large space (for then we may get out of the 
plain), there is between these extremes an extent of 
order. The discovery of this order redeems so much 
from the territory of ignorance, and we apply our 
geometrical rules to it at once. 

Very similar to this is the case with Probability. 
When we get into statistics (supposing that sufficient 
scope is afforded for them) we step, as it were, from 
the rough hill-side on to the level of the plain. Here 
too we may for long have observed no regularity; but 
why? because we had kept our eyes too much fixed 
on the narrow spot where we were standing. The 
first distant view we thought of taking would suggest 



SECT. 16.] LOGIC OF CHANCB. 21 

the idea of uniformity, and the firat measurements 
we made would verify it. In each case, in the midst 
of the apparent chaos of nature — a chaos however 
having its own rules, and in which strict inferences 
of other kinds might be drawn — ^we have found a cer- 
tain portion of ground clearly separated off by its 
mathematical character ; in the former the principles 
of geometry come to our help, and in the latter those 
of arithmetic. 

§ 16. The reader's attention is especially directed 
to the following considerations, to which from their 
importance we shall have to be continually referring. 

1. Let it be observed that experience alone can 
determine the extent over which our mathematics will 
apply. This extent is in no other sense a matter of 
demonstration. We must look for ourselves to see 
where the valley begins and ends, we must tinist to 
obsei'vation to determine how far statistical regularity 
can be reckoned upon. We may make of course what 
use we can of the principles of inductive reasoning and 
analogy to aid us; but our conclusions ultimately rest 
entirely on experience, and are subject at any time to 
be tested and revised by specific experience. 

2. In the next place it must be remarked that, 
although the natural phenomena themselves are of 
limited extent, the mathematics we apply to them 
recognize no limitation. The bottom of the valley 
coincides with a finite portion of a certain horizontal 



22 LOGIC OP CHANCE. [CHAP. I, 

plane, but that plane, of course, may be conceived as 
extending indefinitely (retaining all its properties) in 
every direction. Now in applying the properties of the 
plane to measure the valley, or to make other infer- 
ences about it, we must be very carefnl not to txes-. 
pass beyond the limits which experience will justify; 
in other words, to take no more of the infinite plane 
than is wanted for the finite valley. In a case like 
this, where we can form a visual representation of an 
object, we are scarcely likely to fall into such a mis- 
take; the distinctness with which such representa- 
tions can be made was my reason for choosing the 
illustration; but a precisely similar fallacy is of per- 
petual occurrence in Probability. It is known, for 
instance, that about 250 persons annually commit 
suicide in London. Now if any one were to conclude 
from this that they would continue to do so, and that 
therefore all the sanctions of law and religion were 
vain to prevent them, he would be concluding, as it 
seems to me, that the valley must extend indefinitely 
in all directions. All that Probability, or statistics, 
can say is that if 250 commit suicide certain infer- 
ences may be drawn; but within what limits of time 
the number will remain the same it cannot give a 
hint; this must be decided by a far more complex 
process of experience and induction. The same criti- 
cism will apply to a very great number of the infer- 
ences drawn by social and political writers from the 



SECT. 16.] LOGIC OF CHANCE* 23 

statistics tbey have collected. The reader cannot 
have it too forcibly impressed upon him that the pro- 
cess of inference is as follows; — In examining the 
series of statistics which arise out of any of these 
natural uniformities we should generally find that as 
a matter of fact the series tend at length to lose their 
regularity. But this will not suit our purpose. What 
we do therefore (mde § 14) is to make a substitution, 
and employ instead a series which shall be regular 
throughout. It is by this substituted series that we 
do in reality make our inferences, just as in our illus- 
tration the real basis of our calculations is the ima- 
ginary plane. The validity of the inferences obtained 
clearly depends upon there being a close agreement 
between the substituted series and the real one; when- 
ever the two diverge at all widely we are liable to 
fall into error. The very fact of the series we employ 
being so unlimited in its uniformity makes it more 
enticing to remain in it; an occasional appeal to 
experience therefore is necessary to test and control 
our conclusions. We use our mathematics, in fact, as 
a sort of rail-road; we qait a toilsome and impracti- 
cable path and are whirled along at our ease, often 
through a dark tunnel of symbols; but we must bear 
in mind that we have to get out again ; if we do not 
keep a sharp look out we may be carried far beyond 
our destination. 

3. A third consideration is that we are not to 



24 LOGIC OF CHANCE. [CHAP. I. 

look for any sudden break between the pbenomena to 
which mathematics respectively can and cannot be 
applied. The distinction between the characteristics 
of the two classes, when these characteristics are in 
perfection, is undoubtedly striking, but they merge 
into one another very gradually. Between the valley 
and the hill-side there will be a sloping mass of debris 
insensibly sinking into the former ; the valley itself 
also will not be perfectly true, and may gradually 
change its leveL So in statistics. We shall, in the 
vast majority of instances, as already remarked, find 
that the numerical proportions, which by their persist- 
ence produce the uniformity, gradually change, and 
this to such an extent that the term uniformity at last 
becomes inappropriate. Hence the limits within which 
we collect our statistics are to a certain extent arbitrary ; 
we must exercise our judgment in deciding where we 
will draw the line and what we will include within it. 
§ 17. For additional clearness it may be worth 
while to point out to the reader what, in such an illus- 
tration as that above, would correspond to the distinc- 
tion between the two kinds of series mentioned in a 
previous section. If, as we went along the valley, 
we found its level gradually change and its surface 
become uneven, it would belong to one of those series 
which poi*sess a variable type; if, on the other hand, 
its level surfew^ continued indefinitely, it would belong 
to one of those which possess a fixed type. 



SECT. 18.] LOGIC OP CHANCE. 25 

§ 18. In the foregoing remarks nothing has been 
said that would imply that the series presented a uni- 
formity in more than one respect. As a matter of fact 
observation may detect any number of such uniformi- 
ties, but as they are all of essentially the same kind 
the theory of the subject is unaffected. Thus, for ex- 
ample, in the succession of throws of a penny, we do 
not merely find that the heads are about as numerous 
as the tails, we find also that * heads twice running' 
occura about as frequently as * tails twice running'; 
the same is the case with all other combinations. In 
the same way, if we were examining the tables of 
mortality of London, we should find not merely that 
about 1400 persons died every week, but that the 
numbers of each sex and every age showed some regu- 
larity, and preserved a definite proportion to one ano- 
ther. As all these different general uniformities pre- 
sent precisely the same characteristics, it is sufficient 
for my present purpose to call attention to the fact of 
their existence. Those who desire further informa- 
tion on this subject will find an abundance of inter- 
esting examples in a well-known work on Probability 
by M. Quetelet. I shall recur to the subject in the 
course of the next chapter. 



CHAPTER II. 

THE SERIES OF PROBABILITY ARE OBTAINED BY 

EXPERIMENT AND NOT A PRIORI, AND 

THEY OCCUR IN GROUPS. 

§ 1. At the point which we have now reached we 
are supposed to be in possession of a series of a certain 
kind, lying at the bottom, as one may say, and which 
is the foundation on which the science of Probability 
is erected. The next two enquiries, which have to be 
made in order, are, how in any particular case such a 
series is to be obtained, and in what mode it is to be 
employed when we have obtained it. 

§ 2. The answer to the former enquiry does not 
seem difficult. Experience is our sole guide. If we 
want to discover what is in reality a series of things^ 
not a series of our own conceptions, we must appeal to 
the things themselves to obtain it, for we shall not find 
any help elsewhere. We cannot tell how many per- 
sons will be bom or die in a year, or how many houses 
will be burned or ships wrecked, without actually 
counting them. In saying that experience is the sole 
guide, I mean of course experience supplemented by 
the aids that Inductive Logic can afford. When, for 
instance, we have found the series which represents the 



SECT. 3.] LOGIC OF CHANCBL 27 

numbers of persons who die in successive jears, wo 
have no hesitation in extending it some way into the 
future as well as the past. The justification of this 
procedure must be found in the ordinary canons of 
Induction. As a separate discussion will be given 
upon the connection between Probability and Indue* 
tion, no more need be said on this subject here; but 
nothing will be found at variance with the assertion 
just made, that the series we employ are obtained 
simply from experience. 

§ 3. If Probability were only ooncemed with the 
kind of events which in practice are commonly made 
subjects of insurance, probably no other view than the 
above would ever have obtained credence. But the 
fact of most of its examples having been chosen from 
such things as dice and cards has infected the whole 
science with an d priori tendency, which has biassed 
the minds of its followers in other ttpplications. 

An opinion prevails that in certain cases we are 
able to determine beforehand what the series will be, 
and this with such certainty that the real basis of our 
calculation is not the series itself, but some d priori 
conditions on which the series depends. As I con- 
sider this opinion to be erroneous in fact, and likely 
to cause confusion in the theory of the subject, I will 
proceed to a detailed examination of it; first in its 
stronghold of games of chance, and then in some of its 
other places of occasional resort. From the celebrity 



28 LOGIC OF CHANCE. [CHAP. II, 

of the writers who have maintained this view it could 
not be passed over in silence. 

§ 4. Let us take a very simple example, that of 
tossing up a penny. Suppose that I am contemplat- 
ing beforehand a succession of two throws ; I clearly 
know that the only possible events are* h.h. h.t. t.h. 
T.T. So much is quite certain. We are moreover 
tolerably well convinced from experience that these 
events occur, in the long run, about equally often. 
This is admitted on all hands. But on the view 
against which I am contending it is asserted that we 
might have known the fact beforehand by principles 
which are applicable to an indefinite number of other 
and more complex cases. The form in which this 
view would generally be advanced is, that we are 
enabled to know beforehand that the four throws 
above mentioned are equaUy likely. This is asserted 
by almost every writer on the subject. To this I 
would ask. What is the meaning of the expression 
* equally likely'] To such a question I think but two 
forms of reply are possible. The one of these would 
seek an explanation in the state of mind of the ob- 
server, the other would seek it in some charactei*istic 
of the things observed. (1) It might, for instance, be 

* To those who are not acquainted with the common nota- 
tion of mathematical works, I may say. that H. H. is simply an 
abbreviated way of stating that the two successive throws of the 
penny give head ; and the same with the other symbols. 



SECT. 4.] IiOGIC OP CHANCE. 29 

said on the one hand that what is meant is that the 
two events contemplated are equally easy to imagine, 
or, more accurately, that our expectation or belief in 
their occurrence is equal. We could hardly be con- 
tent with this reply, for the further enquiry would 
then be urged, On what grounds do we believe this 1 
What arc the chamcteristics of events of which our 
expectation is equal? If we consented to give an 
answer to this further enquiry we should be led, I 
think, to the second foim of reply to be noticed directly; 
if we did not consent we should be admitting that 
Probability was only a portion of Psychology, con- 
fined therefore to considering states of mind and not 
the external events to which they referred. We 
should be ceasing to make it a science of inference 
about things. (2) In the other form of reply the ex- 
jjlanation of the phrase in question would be sought, 
not in a state of mind but in a quality of the things 
contemplated. It might give the following as the 
meaning, viz. that the events really would occur with 
equal frequency in the long run. The ground of this 
assertion would probably be found in past experience, 
and it would doubtless be impossible so to fi*ame the 
answer as to exclude the notion of our belief altoge- 
ther. But still there is a broad distinction between 
seeking the equality in the amount of our belief, as 
bi'fore, and in the frequency of occurrence of the 
events, as here; I am convinced that this second form 



30 LOGIC OP CHANCE. [CHAP. H. 

of reply is the correct one, but it may easily be shewn 
that it is tantamount to abandoning the ct priori the- 
ory, for it involves the admission that the real starting 
point of Probability is a sequence of events. 

§ 5. For can the assertion in this reply be made 
h priori ? Those who say it can, have never, I think^ 
fairly faced the difficulties which meet them. For 
the moment we begin to enquire seriously whether 
the penny will really do what is expected of it, we 
shall find that restrictions have to be introduced. In 
the first place the penny must be an ideal one, with 
its sides equal and fair. This restriction is perfectly 
intelligible; the study of solid geometry has enabled 
me to idealize a penny into a circular or cylindrical 
lamina. This restriction is always admitted, but it is 
the only one that is admitted, although it is only one 
condition out of several that combine towards the 
result. But this is not sufficient; there are ether 
conditions. We must also idealize the * randomness' 
of the throwing of the penny, which is a process that 
one feels rather at a loss to know how to set about 
performing. This is no idle subtlety; for will it be 
asserted that the heads and tails would get their &ir 
chances supposing that, in the act of throwing, I were 
always to start the same side uppermost? Scarcely, 
I think; the difference, slight as it is, might become 
at last appreciable. Or if I persisted in starting 
with the two sides upwards alternately, would the 



8ECT. 6.] LOGIC OF CHANCE. 31 

long repetitions of the same side get their fair chance? 
Perhaps it will be replied that we are to think no- 
thing at all about these matters, and all will come 
right. It may, and doubtless will, but this is falling 
back upon experience. For suppose, lastly, that the 
circumstances of nature or my bodily or mental con- 
stitution were such that the same side always i8 started 
uppermost; well, it will be replied, it would not then 
be a fair trial. I am convinced that if we press in 
this way for an answer to such enquiries, we shall 
find that these tacit restrictions on the d priori plan 
are really nothing else than a mode of securing an 
experimental result. They are only a way of saying, 
Let a series of actions be performed in such a way as 
to secure a sequence of a particular kind, viz. of the 
kind described in the last chapter. 

The remarks above made will apply, of course, to 
most of the other common examples of chance; the 
throwing of dice, drawing of cards, of balls from bags, 
&C. In all these cases, if we scrutinize our language 
carefully, we shall find that the supposed ct priori 
mode of stating the problem is little else than a com- 
pendious way of saying, Let means be taken for ob- 
taining a given result. Since, then, it is upon this 
result that our inferences ultimately rest, it seems to 
me simpler and more philosophical to appeal to it at 
once as the groundwork of our science* 

§ 6. It will be seen, by this time, how very nar- 



82 LOGIC OF CHAKOE. [CHAP. II. 

row is the range of cases which the so-called ^ priori 
plan can be supposed to embrace. It is confined to 
games of chance, and can only be introduced there 
bj the aid of many tacit restrictions. This alone 
would be conclusive against the theory of the subject 
being based upon it. The experimental plan, on the 
other hand, is of universal application. It would in- 
clude the ordinary problems of games of chance, as 
well as those where the dice are loaded and the pence 
are not ideal, and also the indefinitely numerous ap- 
plications of statintics to social phenomena and the 
&cts of inanimate nature. 

§ 7. I quite admit the advantages of the ^ priori 
plan where it can be used. Without it chance problems 
could scarcely be set in examinations, and the science 
would be deprived of a certain neatness and inde- 
pendence which the common mode of treatment con- 
fers upon it. Moreover, in many cases it would be a 
real hardship to be debarred from appealing to it. We 
are often enabled, by geometrical and other considera- 
tions, to ascertain with tolerable accuracy what kind 
of a sequence of events we may look for, at a time 
when we are either without specific experience or 
should find it tedious to obtain it. Against this as a 
practical measure not a word of objection can be 
raised ; I am only contending that it is not the sim- 
plest and most consistent way of studying the theory 
of the subject. We may use an artifice for obtaining 



1 



SECT. 8.] LOGIC OP CHANGE. 33 

the series, but it would be a great mistake to take 
anything but the series as the foundation of our rules. 

§ 8. The €t priori theory, in the form examined 
above, could scarcely have intruded itself into any 
other examples than those drawn from games of chance. 
But a doctrine, which is in reality little else than the 
same theory under a slightly disguised form, is very 
prevalent, and has been applied to truths of the most 
purely experimental character. This doctrine will be 
best introduced by a quotation from Laplace. After 
speaking of the irregularity and uncertainty of nature 
as it appears at first sight, he goes on to remark that 
when we look closer we begin to detect* ** a striking 
regularity which seems to arise from design, and 
which some have considered a proof of Providence. 
But, on reflection, it is soon perceived that this regu- 
larity is nothing but the development of the respective 
probabilities of the simple events, which ought to 
occur more frequently according as they are more 
probable." 

Now if this remark had been made about the sue- 
eession of heads and tails in the throwing up of a 
penny, it would have been intelligible, though, as I 
have endeavoured to show, not philosophically correct. 
It would simply mean this : that the constitution of 
the body was such that we could anticipate what the 

• Translated from Laplace, Essai Pkilosopkique, p. 74. 

3 



34: LOGIC OP CHANCE. [cHAP. H. 

result would be when it was treated in a certain way, 
and that experience, in the long run, would justify 
our anticipation. But applied as it is in a more 
general form to the facts of nature it seems altogether 
unmeaning. We will test it by taking what is per- 
haps one of the strongest conceivable instances in its 
support. Amidst the iiTegularity of individual births, 
we find that in the long run the male children are ta 
the female in about the proportion of 106 to 100. 
Now when we are told that there is nothing in this 
but the "development of their respective probabili- 
ties," what is there in this sentence but a somewhat 
pretentious restatement of the fact just asserted ] The 
probability is nothing but that proportion, and is de- 
rived from the statistics alone; in the above remark 
the attempt seems made to invert this process, and to 
derive the sequence of events from a numerical state- 
ment about the very events themselves. 

§ 9. It will be said perhaps that by the probability 
above mentioned is meant, not the mere numerical 
proportion between the births, but some fact in our 
constitution upon which this proportion depends; that 
just as there was a relation of equality between the 
two sides of the penny, so there may be something in 
our bodies in the proportion of 106 to 100 which pro- 
duces the statistical result. 

When this something, whatever it might be, was 
discovered, the observed numbers might be supposed 



SECT. 10.] LOGIC OP CHANCE, 35 

capable of being determined beforehand*. Even if 
this were the case it is forgotten that there must be, 
in combination with such a cause, other conditions in 
order to produce the ultimate result ; just as the ran- 
domness of the throw was combined with the equality 
of the two sides of the penny. This is quite suffi*- 
cient to prevent us from obtaining anything which 
could strictly be called the objective probability of 
the events. So far as can be seen this example pro- 
bably offers no exception to the general rule of a 
slowly but irregularly varying type. Even here there- 
fore where the conceptions involved in the ^ priori 
theory seem most appropriate, they will be found, I 
think, to fail on examination. 

§ 10. The reader who is familiar with Probability 
is of course acquainted with the celebrated theorem of 
Bernoulli. This theorem, of which the examples just 
adduced are meuely particular cases, is generally ex- 
pressed somewhat as follows : — ^that in the long run 
all events will tend to occur with a frequency propor- 
tional to their objective probabilities. With the ma- 

• Physiologists, I believe, are of opinion that the relative 
ages of the father and mother have something to do with the sex 
of the offspring. If this be so, it completely bears out what is 
said above, for it introduces into the consideration an element 
dependent in some degree upon the civilization and sentiments 
of any particular age, an element which may possess any degree 
of irregularity. As a matter of fact, moreover, the proportion 
of io6 to I CO given above, does not seem by any means uni- 
versal iu all countries or at all times. 

3—2 



36 I/)GIO OP CHANCE. [CHAP. II, 

thematical proof of this theorem I have nothing to 
do here; nor, if there is any value in the foregoing 
criticism, need we trouble ourselves about it, for in. 
that case the basis on which the mathematics rest is 
faulty, owing to the fact of there really being nothing 
which we can call the objective probability. 

This theorem of Bernoulli seems to me one of the 
last remaining relics of Kealism, which afber being 
banished elsewhere still manages to linger in the 
remote province of Probability. It is an illustration, 
of the inveterate tendency to objectify our conceptions 
even in cases where the conceptions had no right to 
exist at all. A uniformity is observed ; sometimes, as 
in games of chance, it is found to be so connected 
with the physical constitution of the bodies employed 
as to be capable of being inferred beforehand, though 
even here the connection is by no means so necessary 
as is commonly supposed; this constitution is then 
converted into an "objective probability," supposed 
to develop somehow into the sequence which exhibits 
the uniformity. Finally, this very questionable ob- 
jective probability is assumed to exist, with the same 
faculty of development, in all the cases in which 
uniformity is observed, however little resemblance 
there may be between these and games of chance. 

§ 11. How utterly inappropriate any such concep- 
tion is to most of the cases in which we find statis- 
tical uniformity, will be obvious on a moment's consi- 



SECT. 12.] LOGIC OP CHANCE. S7 

deratioiL The observed phenomena are generally the 
product, in these cases, of very numerous and compli- 
cated antecedents. The number of crimes, e. g. an- 
nually committed, is a function of the morality of the 
people, their social condition, and the vigilance of the 
police, each of these elements being in itself almost 
infinitely complex. Kow as a result of all these 
agencies, there is some degree of uniformity, but what 
I have called above the chcmge of type in it is most 
marked. The annual numbers fluctuate in a way 
which, however it may depend upon causes, shows 
none of the permanent uniformity of games of chance. 
This, combined with the obvious arbitrariness of sin- 
gling out some only from the many antecedents which 
produced the regularity, would have been quite suf- 
ficient to prevent any one from assuming the exist- 
ence of any objective probability here unless he had 
been predisposed to believe in it. 

§ 12. I have been thus minute in the criticism 
of the above doctrine, because I think that, besides its 
intrinsic eiTor, it has a strong tendency to obscure the 
due perception of two very important positive truths 
already mentioned; firstly, the gradual change of 
type, and secondly, the distinction between the actual 
series about which we reason and the substituted 
series we employ in reasoning about it. 

Its bearing upon the first is obvious. The doctrine 
of an objective probability almost necessarily presup- 



38 LOGIC OP CHANCE. [CHAP. II. 

poses a fixed type. It seems merely the realistic 
doctrine of an ideal something which is perpetually 
striving, and gradually, though never perfectly, suc- 
ceeding in realising itself in nature. Against this 
the view of a changeable type is, of course, distinctly 
antagonistic; still more so when, as I maintain, the 
type not merely changes, but has in many cases an 
actual origin and conclusion. I cannot help thinking 
also that the very common practice of carrying on 
statistical calculations almost indefinitely springs from, 
and is tainted by, the same error. If the type were 
fixed we could not have too many statistics, but if it 
vary, our extra labour may be worse than wasted. 
The danger of stopping too soon is easily seen, but in 
avoiding it we must not fall into that of going on too 
long. 

The other caution is equally important I have 
called attention to it already in the first chapter, and 
shall have to recur to it again at such length that I 
only state it here. The student cannot have too 
earnestly impressed upon him the fact that there is 
such a distinction. 

§ 13. A possible objection may be raised here 
against calling the above results purely experimental^ 
on the ground that it is the very essence of an average 
to diminish differences; that the arithmetical process 
of obtaining it insures this, whatever may be the pro- 
perties of the things themselves. For instance, let 



SECT. 13.] LOGIC OP CHANCE. 39 

there be a party of ten men, of whom one or two are 
tall and one or two shorty and take the average 
height of any five of them. Since this number can- 
not be made up of tall men only, or short men only, 
it stands to reason that the averages cannot differ so 
much amongst themselves as the single measures may. 
Is not then the equalizing process, it may be asked, 
which is observable on increasing the range of our 
observations, one which can be shown to follow from 
necessary rules of arithmetic, and one therefore which 
might be asserted ^ priori f Whatever apparent force 
there may be in this argument arises principally from 
the arbitrary limitations of the particular example 
given above, in which the number chosen was so large 
a proportion of the total as to exclude the possibility 
of only extreme cases being contained within it. 
Let us take an instance in which the number selected 
is by comparison small ; suppose a succession of deaths, 
and examine the average duration of any ten lives. 
No possible reason can be assigned why any ten in- 
fants should not all live to the age of 100, or all die 
on the day after they were bom. We can simply 
state the fact, discovered and proved by experience, 
that Isuch averages are more nearly equal than the 
individual lives which compose them. It is doubtless 
quite true that if we take a given limited number of 
examples and divide them up into groups, we shall 
£nd that the averages of these groups present far 



40 LOGIC OP CHANCE. [CHAP. II, 

more of uniformity than the single examples of which 
these averages are composed. If the groups bear anj 
considerable proportion to the whole this is necessarily 
the case. But it is quite a different thing to assert 
this when the groups do not bear more than a small 
proportion to the series out of which they are taken, 
still more so when the numbers in the series are prac- 
tically or even really unlimited. The average height 
of the different batches of fifty men that might be 
selected from a party of eighty must be tolerably 
nearly equal; but there is no such obligation when 
the fifty are selected from one thousand, still less when 
they are selected from twenty millions. This latter 
is the characteristic of things as they present them- 
selves in nature, and it is this which I assert to be 
solely established by experience. 

The distinction may be made plainer by the help 
of another example. Suppose that we are considering 
a succession of throws of a penny. We select a 
limited number, say ten, and we assume it known 
that there are heads and tails amongst them. Both 
these restrictions are requisite before we can infer 
anything necessarily, in other words h priori; but 
let us see what is the real extent of this necessary 
inference. It seems simply this, that whereas when 
we select but a few of the ten we cannot tell but 
what they may be all heads or all tails, w«e may know 
for certain that by taking a large enough proportion 



SECT. 14.] LOGIC OP CHANCE. 41 

of the whole we escape such an extreme inequality, 
and must find specimens of both head and tail. But 
this is an extremely different thing from being able 
to assert that as we continue to take more and more 
out of the indefinite series of possible throws, we 
shall find the relative proportions of heads and tails 
gradually approaching towards equality. This indeed 
we are also able to assert, but from very different 
reasons. It can only be done by inductive extension 
from actual experience. 

§ 14. The distinction which has just been described 
and illustrated corresponds in part to one which M. 
Quetelet has drawn between what he calls arithmetical 
averages and means, and to which he attaches ex- 
treme importance. It certainly is a very important 
distinction; but as his description of it, though inter- 
esting and very serviceable for practical purposes, 
seems to me to be involved in serious confusion in 
regard to the theory of the subject, I will quote and 
examine it at some length. It will serve as a good 
introduction to a fuller investigation of the nature 
of the series with which we are conceraed in Proba- 
bility than could be attempted in the last chapter. 

He says*, " In measuring the height of a building 
twenty times in succession, I may not perhaps twice 
find the same identical valae. However, it may be 

* Quetelet On Prdbo^iliUeay by O. G. Downes, p. 4a, 



42 LOGIC OP CHANCE. [CHAP. II. 

ooDceived that tlie building has a determinate height; 
and if I have not exactly estimated it, in any one of 
the operations I have made to discover it, it is because 
these operations are liable to some uncertainty. I 
content myself, then, by taking the average of all my 
results as the true height sought. The limits, greater 
or smaller, depend on my skill or want of skill, and 
on the exactness of the instruments which I have 
used I may employ the calculation of the mean in. 
another sense. I wish to give an idea of the height 
of the houses in a certain street. The height of each 
of them must be taken, and the sum of the observed 
heights must be divided by the number of houses. 
The mean will not represent the height of any par- 
ticular one ; but it will assist in showing their height 
in general; and the limits, greater or smaller, will 
depend on the diversity of houses. There is between 
these two examples a very remarkable difference, 
which perhaps may not have been seen at first glance, 
but which is nevertheless of great importance. In 
the first, the mean represents a thing really in exist- 
ence; in the second, it gives, in the form of an abs- 
tract number, a general idea of many things essen- 
tially different, although homogeneous. In another 
view (and this point is important) the numbers which 
have contributed to form the mean in the two exam- 
ples present themselves in very different manners. In 
the second example, they are bound to one another by 



SECT. 15.] LOGIC OP CHAXCE. 43 

no law of continuity; while in the first, as we shall 
soon have occasion to see, the determinations of the 
heights, although faulty, range themselves on each 
side of the mean with so great a regularity that their 
values might be predicted, if the limits within which 
they are comprised were given." 

§ 15. The first remark I have to make upon this is 
to point out how the realistic doctrine, already alluded 
to, begins to creep in here. In one of the examples 
chosen there is a real and fixed value, viz. the height 
of the building, towards which the measurements 
strive to attain. But the same assumption is soon 
extended to cases in which there is no such fixed 
value. Thus for the measurements of the height of 
a house, are afterwards introduced those of the height 
of a man. This is all very well ; but when, on com- 
paring the group which these attempted measures 
compose, we find that they correspond roughly to the 
actual heights of a large number of different men 
taken at random, and go on to assume, as M. Quetelet 
does, that these actual heights must be in some way 
modelled on a type common to all : what is this but 
the reappearance of the realistic doctrine ) 

It might be true, as a matter of fact, that the two 
series of heights, viz. those of real men and those 
obtained by imperfect calculations from one model 
should be about the same; though, as Sir John Her- 
schel has pointed out, considerable violence has in 



44 LOGIC OP CHANCE. [cHAP. H. 

reality to be done to the real heights before they can 
be forced into reasonable compliance with such a sup- 
position. But even then there would be no meaning 
in asserting the real existence of the type or mean. 

It is almost needless to point out that nothing 
which is said here is opposed to the conception of a 
type as employed by Comparative Anatomists. With 
them it is nothing more, I apprehend, than a state- 
ment of resemblances which are actually found to sub- 
sist between different species. If any additional hypo- 
thesis be intended, I presume it would be one of a 
causal nature as to the process by which these species 
and their varieties had been produced. This is quite 
different from the existence of a real type in the sense 
which M. Quetelet seems to contemplate. 

§ 16. But there is a far more serious error, or, 
perhaps one should say, a far more serious confusion 
involved in the passage which has been quoted. A very 
important experimental result is slurred over in it; 
the different forms which this result may assume being 
not only treated as if they were necessarily alike in 
all examples, but the assumption being made that thia 
form might have been ascertained deductively. 

For clearness of conception let us first separate off 
the arithmetical average. This, as I have shown 
already, is entirely an abstraction of our own, and 
refers to a limited number of things or observations. 
If it embrace a large proportion of the whole numberi 



SECT. 17.] LOGIC OF CHANCE. 45 

we may know for certain, except under special cir- 
cumstances, that different averages so taken cannot 
differ from one another so much as the different single 
things may. 

But now leaving this limited number, let us go on 
to contemplate the unlimited series from which the 
selection embraced by the average might be conceived 
to be taken. I may remind the reader that it does 
not matter, for our present pui-pose, of what sort of 
things this series is supposed to consist. The indivi- 
duals which compose it may be presented to us in 
nature, or they may be products of our own in the 
form of attempted and erroneous measurements. In 
the latter case there is of course a real fixed value to 
which the measurements strive to attain ; in the 
former case also it is often assumed that there must 
be something existing which occupies a similar posi- 
tion to the real and fixed value. I think this assump- 
tion is unwarranted, but in any case it is an inference 
with which we are not at present concerned. It is 
the series itself only which we are going to examine. 

§ 17. When we investigate the different series 
which M. Quetelet has given as illustrative of what he 
terms a mean, it is true that we may detect in them a 
characteristic which was not brought under notice in 
our description of the distinctive series of Probability 
in the last chapter. But it is a characteristic, I ap- 
prehend, which does not in the slightest degree affect 



4iS LOGIC OP CHANCE. [CHAP. IL 

the principles of the subject. It seems to me indeed 
to be one, which though inseparably connected with 
questions of Probability, is strictly speaking discon- 
nected from the theory of the science. 

The difference in fact between such examples as 
he mentions, and those which have hitherto occxipied 
our attention, consists in the fact of the latter having 
what we have called a single type, and the former being 
composed of what might be called a group of types 
combined in a certain way. If, for instance, we exa- 
mine a number of successive deaths we find about as 
mauy of them are male as female. But we may dis- 
cover a great many more uniformities in them than 
this. Let us take a batch of ten. This batch may be 
made up in a variety of ways, according to the pro- 
portion of male and female deaths which compose it; 
If we repeat the process often enough we shall find 
that in a certain number of cases, and these the most 
numerous, the ten will be made up of five of each. In 
a somewhat smaller number it will be made up of six 
men and four women, in a still smaller number of seven 
men and three women, and so on. Similar proportions 
will hold if we substitute women for men. Here we 
have the notion of a mean, and all other examples of 
means will be found to resemble this in their essential 
characteristics. On analysis it is seen to be decompo- 
sable into a group of uniformities, each separate unifor- 
mity possessing the features characteristic of our science^ 



SECT. 18.] LOGIC OP CHANCE. 47 

and the members of the group being united together 
by some de&iite law. This law of combination of the 
groups must be determined by experience, just like 
any of the features in the separate groups themselves. 
Of this law of combination I shall say no more at pre- 
sent, as I do not consider that it admits of any very 
accurate determination, and in any case it does not 
affect the general theory of our subject, which depends 
upon each separate uniformity preserving the essential 
feature belonging to examples in Probability. 

§ 18. We now see clearly the distinction between 
the average and the mean. The former is simply the 
arithmetical average obtained in the ordinary way, and 
therefore referring necessarily to a limited number of 
things j it is moreover a pure abstraction of our own. 
If however these things are supposed to be a portion 
of an indefinite series of the well-known kind, they 
become fitting subjects for our science. If moreover 
several such series, whether composed of different sets 
of things or of the same things differently arranged, 
group themselves into symmetry, then we have the 
notion of a mean. As a matter of fact experience proves 
that almost all natural objects, when closely examined, 
are found to arrange themselves in groups of uniform- 
ities, and that there is throughout nature a tolerably 
regular law in accordance with which these groups are 
composed. Suppose a large number of persons bom 
at the same time* If we examine the length of their 



48 LOGIC OP CHANCE. [CHAP. II. 

lives we shall find that they do not die, as it would 
be said, at random. A certain proportion will die in 
their first year, a certain proportion again in their 
second year, and so on. But these different propor- 
tions will themselves be found to be united together 
in a certain tolerably uniform manner. Again, if we 
examine a succession of deaths of men and women, we 
shall find, not merely that there are, on the average, 
about as many deaths of one sex as of the other, as 
many repetitions of two male deaths as of two female, 
and so on : we shall find that the numbers of these dif- 
ferent repetitions are themselves grouped together in 
an orderly way, which becomes most marked when we 
take a very large number of examples. Take another 
example, of an entirely difibrent kind. Suppose that 
a man had been practising for some time with a pistol 
at a target. The shots will be found to cluster about 
the centre, becoming less frequent the greater their 
distance from the centre. There will not only be 
about as many on one side as on the other at each 
different distance from the centre, but these numbers 
will be found to be graduated in an orderly and regular 
way in the long run. 

Now in all these cases, any number more of which 
might be adduced, we observe the same characteristic. 
Not only do they present a collection of uniformities 
of the kind described so fully in our last chapter, but 
this collection is grouped in an orderly and regular 



SECT. 19.] LOGIC OF CHANCE. 49 

way. Hence if we have any means of finding out wliat 
is the law of connection of these groups, and can feel 
sure that there is but otie law, we should then be able, 
from a fragment of one of these uniformities, to infer 
the whole of it, without a fresh appeal to experience. 

In the foregoing remarks I have striven to draw 
out as clearly as I can what I conceive M. Quetelet to 
have been aiming at. But his description seems to 
me so confused that I may have &iled to understand 
him. Be this as it may, the account given above 
appears to me to be a correct description of the facts 
as they are presented in nature. 

§ 19. Let us now see what conclusions are drawn 
from the facts. These conclusions appear to be resolv- 
able into a generalization, and an inference grounded 
upon that generalization. The generalization is that 
there is, if not one type for all the groups described 
above, at least one general principle upon which these 
types are founded. The inference is that since in cer- 
tain cases there is undoubtedly a real objective thing at 
which the elements which go to make up the group 
are aimed, there must be something corresponding to 
this in all the cases. Of this inference I have already 
said all that seemed requisite; let us therefore turn for 
a moment to examine the generalization. 

It is strange that this should ever have found ac- 
ceptance, except as a rough approximation. Even M. 
Quetelet's. own example, that of the heights of man- 

4 



50 LOGIC OP CHANCE. [CHAP. II. 

kind, can only be brought into apparent agreement 
with his formula by the violent expedient of rejecting 
the extremes at both ends of the list on the plea that 
they are 'monstrosities.' This rejection is perfectly- 
arbitrary, for these dwarfs and giants are bom into 
the world like their better-shaped brethren and have 
precisely the same right to find themselves included 
in the formula. And even, by the help of this expe- 
dient, as has been mentioned already, the examples fit 
in but very lamely. 

But there does not seem to be any need to appeal 
to particular examples, for there is the following ob- 
vious and notorious failure in the generalization. In 
some of the series which it attempts to embrace, 
e. g. successions of heads and tails in the throws of a 
penny, there is no finite limit to the individual fluc- 
tuations; in other cases, including almost all the a}>- 
plications of Probability to natural phenomena, there 
are such limits. This of course is well known, the 
reply generally being that these extreme deviations 
beyond a certain point are so excessively rare that no 
practical error is produced by assuming them to exist 
where in reality they do not. No practical error 
perhaps, but still enough to vitiate the theory, for the 
extreme cases are in every instance produced by the 
same agencies as those which produce all the interme- 
diate ones; we cannot therefore conceive any alteration 
in the formula when it produced the one without in- 



SECT. 20.] LOGIC OF CHANCa 51 

juring its integrity for the remainder. We cannot in- 
troduce hypothetical extreme vaiiations, however scarce 
we make them, without making a consequent change^ 
slight though it may be, in all the intermediate ones. 

§ 20. If it be urged that the employment of one 
general rule (that familiar to mathematicians under 
the name of Least Squares) under very various cir- 
cumstances, as for example in astronomical observa- 
tions, &c, proves the validity of the generalization in 
question, I should reply that few persons have reflected 
upon the immense extent of experience that would be 
requisite for the purpose. To show that one rule 
gives correct results in the long run is not sufficient; 
it is necessary to show that no other rule would do as 
much. A very gi'eat number of observations would 
probably be necessary to show that of two formulae, 
both of which recognized the principle that wide de- 
viations from the mark were less common than small 
ones, only the one which admits a particular law for 
the diminution of these deviations was capable of 
being used with correct results. Only by such an 
investigation, I think, could it be ascertained that all 
the series which occur in nature do approximately 
conform to a common type, so as to be capable of 
being genei*alized under a common formula. They 
certainly do not accurately conform, for the reason 
given already, viz. that some of them admit infinite 
fluctuations whilst the majority do not. 

4—2 



52 LOGIC OF CHANCE. [cHAP. II. 

§ 21. If the reader will carefiilly study the fol- 
lowing example, one well known to mathematicians 
as the Petersburg Problem, I think it will serve to 
illustrate the three following considerations, at least, 
out of those which have occupied our attention in this 
and the preceding chapter : — (1) The distinction be- 
tween the actual series of observation and the sub- 
stituted one of calculation. (2) The fact that this 
latter is not hampered by the limits which experience 
imposes upon the former, and is therefore indefinite 
in the extent of its potential range. (3) That cer- 
tain series take advantage of this indefinite range to 
keep on producing individuals in it whose deviation 
from the average has no finite limits whatever. When 
rightly viewed it is a very simple problem, but it 
has given rise to a great deal of confusion and per- 
plexity. 

The case is this :-^-a penny is tossed up ; if it 
gives head I receive two shillings ; if heads twice 
running four shillings ; if heads three times running 
eight shillings, and so on ; the amount doubling every 
time. In a word, however many times head may be 
given in succession, the number of shillings I may 
claim is found by multiplying two by itself that 
number of times. Here then is a series formed by a 
succession of throws. I will assume, — what most 
persons will consider to admit of demonstration, and 
what certainly experience confirms within consider- 



SECT. 21.] LOGIC OP CHANCE. 53 

able limits, — that the rarity of these 'runs' of the 
same face is in direct proportion to the amount I 
receive for them when they do occur* In other 
words, if we regard the occasions on which I receive 
payments, 1 shall find that every other time I get 
two shillings, once in four times I get four shillings, 
once in eight times eight shillings, and so on with- 
out end. The question is then asked, what ought I 
to pay for this privilege? At the risk of a slight 
anticipation of the next chapter, I may assume that 
this is equivalent to asking, what amount paid each 
time would, on the average, leave me neither winner 
nor loser 1 in other words, what is the average amount 
that I should receive on the above terms? Theory 
pronounces that I ought to give an infinite sum, 
that no finiiie sum, however great, would be an ade- 
quate equivalent. And this seems quite intelligible; 
there is a series of indefinite length before me, and 
the longer T continue to work it the richer are my 
returns, aod this without any limit whatever. It is 
true that the very rich hauls are extremely rare, 
but still they do come, and, when they come, they 
make it up by their greater richness. On every 
occasion on which people- have devoted themselves 
to the pursuit in question, they made acquaintance, 
of coui'se, with but a limited portion of this series ; 
but the series on which we base our calculation (which 
I have aboY^ described as the substituted or ideal 



54 LOGIC OP CHANCE. [cHAP. H. 

series) is tinlimited, and the inferences we have drawn 
are in perfect accordance with this. 

The common Jform of objection to this is given in 
the reply, that so far from paying an infinite sum 
no sensible man would give £50 for such a chance. 
Probably not, because no man would see enough oi 
the series to make it worth his while. What most 
persons form their practical opinion upon, is such 
small portions of the series as they have actually seen 
or can reasonably expect. Now in any such portion, 
say of 100 throws, the longest succession of heads 
would not amount on the average to more than six or 
seven. This is observed, but it is forgotten that the 
formula which produced these, would, if it had greater 
scope, keep on producing better ones. Hence it arises 
that some persons are perplexed, because the conduct 
they would adopt in reference to the curtailed portion 
of the series which they practically meet with does 
not find its justification in inferences which are 
avowedly based on the series in the completeness of 
its infinitude. I shall have occasion to refer to the 
problem again in a future chapter. 

§ 22. The results obtained in this chapter may be 
summed up as follows : — ^We have extended the con- 
ception of a series obtained in the last chapter; for we 
have found that nature presents these series to us in 
groups. These groups are divisible into two classes, 
which offer a marked contrast in the extreme cases 



SECT. 22.] LOGIC OF CHANCE. 55 

whieh they embrace, but offer a general similarity in 
other respects; though it would be hard to prove the 
existence of anything more between them than this gene- 
ral similarity. This similarity has given occasion, firstly 
to a sort of realistic inference, which seems almost un- 
meaning; and secondly to the belief that some of the 
series can be obtained deductively, which I have at- 
tempted to disprova All that we can safely say about 
obtaining them is this, that by Inductive extension we 
may often, from a fragment of one of these series, infer 
the remainder of the series, and thence infer the other 
series which go to make up the group. A full dis- 
cussion of the connection between Induction and Pro- 
bability is reserved for a future chapter. 



1 



CHAPTER III. 

GRADATIONS OF BELIEF. 

§ 1. Having now obtained a clear conception of a 
certain kind of series, the next enqtdry is, What is to be 
done with this series 1 How is it to be employed as a 
means of making inferences ? The first step that we 
are now about to take might be described as one from 
the objective to the subjective, from the things them- 
selves to the state of our minds in contemplating them. 
The reader should observe that a substitution has 
already been made as a first stage towards biinging 
the things into a shape fit for calculation. This sub- 
stitution, as described in the last chapter, is, in a mea- 
sure, a process of idealization^ corresponding, in our 
illustration, to the substitution of an ideal plane for 
the real bottom of the valley. The series we actually 
meet with show a changeable type, and the indivi- 
duals of them will sometimes transgress their licensed 
irregularity. Hence they have to be pruned a little into 
shape, as natural objects always have before they are 
capable of being accurately reasoned about The form 
in which the series emerges is that of a series with a 



SECT. 1.] LOGIC OP CHANCE. 57 

fixed type, and with its unwarranted irregularities 
omitted This imaginary or ideal series is the basis 
of our calculation. 

It must not be supposed that this is at all at 
variance with the assertion preyiouslj made, that 
Probability is a science of inference about real things ; 
it is only by a substitution of the above kind that 
we are enabled to reason about the things. In na- 
ture they present themselves in a form not rigor- 
ously scientific, just as the bottom of the valley 
did ; and as we had there to interpose an imaginary 
plane, so we have here to introduce an imaginary 
series. The only condition to be fulfilled in either 
case is, that the substitution is to be as little arbi- 
trary, that is, to vary from the truth as slightly as 
possible. This kind of substitution generally passes 
without notice when natural objects of any kind are 
made subjects of exact science. I direct distinct atten- 
tion to it here simply from the apprehension that want 
of familiarity with the subject-matter might lead some 
readers to suppose that it is, in this case, an excep- 
tional deflection from accuracy in the formal process 
of inference. 

I may remark also that this imaginary series offers 
no countenance whatever to the 'objective probability' 
doctrine criticised in the last chapter. It differs from 
anything contemplated on that hypothesis by the fact 
of its being recognized as a necessary substitution of 



58 LOGIC OP CHANGE. [CHAP. HI. 

our own for the actual series, and to be kept in as 
close conformity with it as possible. It is a mere 
fiction or artifice necessarily resorted to for the pur- 
pose of calculation, and for this purpose only. 

This caution is the more necessary, because in the 
example that I shall select, and which is one of thei" 
most favourite class of examples in this subject, the 
substitution becomes accidentally unnecessary. The 
things may sometimes need no trimming, because in 
the form in which they actually present themselves they 
are almost idealized. It is as if, at the bottom of our 
valley, we came upon a large sheet of ice; we could 
scarcely even imagine a more perfect plane than this. 
In most cases a good deal of alteration is necessary to 
bring the series into shape, but in some — I refer of 
course to games of chance — we find the alterations, 
for all practical purposes, needless. 

§ 2. We start then, from such a series as this, 
npon the enquiry. What kind of inferences can be 
made about it? It may assist the logical reader to 
inform him that our first step will be analogous 
to one class of what are known as immediate in- 
ferences, — inferences, that is, of the type, — ^All men 
are mortal, therefore any particular man or men 
are mortal This case, simple and obvious as it is 
in Logic, requires very careful consideration in Pro- 
bability. 

It is obvious that we must be prepared to form 



SECT. 2.] LOGIC OF CHANCE. 69 

an opinion npon the propriety of taking tbe^step 
involved in such an inference. Hitherto we have 
had as little to do as possible with the iiregular in- 
dividuals; we have regarded them simply as frag- 
ments of a regular series. But we cannot long con- 
tinue to neglect all consideration of them. Even if 
these events in the gross be certain, it is not only in 
the gross that we have to deal with them ; they 
constantly come before us a few at a time or even 
as individuals, and we have to form some opinion 
about them in this state. An Insurance Office, for 
instance, deals with numbers large enough to obviate 
uncertainty, but each of their transactions has another 
party interested in it — ^What has the man who in- 
sures to say to their proceedings ? for to him this 
question becomes an individual one. And even the 
Office itself receives its cases singly, and would there- 
fore like to have as clear views as possible aboiit 
these single cases. Now, the remarks made in the 
last two chapters about the subjects which Pro- 
bability discusses might seem to preclude all en- 
quiries of this kind, for was not ignorance of the 
individual presupposed to such an extent that even 
(as will be seen hereafter) causation might be denied 
without affecting our conclusions 1 The answer to 
this enquiry will require us to turn now to the con- 
sideration of a totally distinct side of the question, 
and one which has not yet come before us. Our 



60 LOOK) OF CHANCE. [CHAP. III. 

best introduction to it will be hj the discussion of 
a special example. 

§ 3. Let a penny be tossed up a very great 
many times ; we may then be supposed to know for 
certain this fact (amongst many others) that in the 
long run head and tail will occur equally often. 
But suppose we consider only a moderate number of 
throws, or fewer still, and so continue limiting the 
number until we come down to three or two, or even 
one) We have as the extreme cases certainty or 
something undistinguishably near it, and utter uncer- 
tainty. Have we not, between these extremes, all 
gradations of belief] There is a large body of writers, 
including some of the most eminent authorities upon 
this subject, who reply that we are distijictly con- 
scious of such a variation of the amount of our be- 
lief, and that this state of our minds can be measured 
and determined with almost the same accuracy as 
the external events to which they refer. The prin- 
cipal mathematical supporter of this view is Professor 
De Morgan, who has insisted strongly upon it in all 
his works on the subject. The clearest exposition of 
his opinions will be found in his Formal Logic, in 
which work he has made the view which we are now 
discussiug the basis of his system. He holds that 
we have a certain amount of belief of every propo- 
sition which may be set before us, an amount which 
in its nature admits of determination, though we may 



SECT. 4.] LOGIC OF CHANCE. 61 

practically find it difficult in any particular case to 
determine it. He considers, in fact, that Probability 
is a sort of sister science to Formal Logic, speaking 
of it in the following words : " I cannot understand 
why the study of the effect, which paHial belief of 
the premises produces with respect to the conclusion, 
should be separated from that of the consequences of 
supposing the former to be absolutely true." In other 
words, there is a science — Formal Logic — which in- 
vestigates the rules according to which one proposi- 
tion can be necessarily inferred from another; cor- 
responding with this there is a science which investi- 
gates the rules according to which the amount of our 
belief of one proposition varies with the amount of 
our belief of other propositions with which it is 
connected. 

§ 4. If this were the opinion of Professor De 
Morgan only, or even of mathematicians generally 
(and I believe that substantially the same opinion is 
adopted by all who have treated the subject mathe- 
matically), it might be objected that their peculiar 
studies had given them a bias towards discovtring 
the distinctions and accuracy of numbers in matters 
into which these qualities are not commonly sup- 
posed to enter. But it must be observed that a 
professed logician, Archbishop Thomson, has to a 
considerable extent adopted the same opinion. In 
his work on the Laws of Thought he gives a distinct 



62 LOGIC OP CHANCE. [CHAP. lH. 

section to the treatment of what he calls * Syllogisms 
of Chance.' He prefaces it with a statement that the 
substance of the section is extracted from the works 
of Professor De Morgan, and others who agree with 
Professor De Morgan ; he also makes a quotation 
from Professor Donkin, with which he seems to agree, 
which declares that the subject matter of the science 
of Probability is * quantity of belief.* I must con- 
fess, with all respect to the Archbishop, that this 
chapter has always appeared to me less acute than 
the rest of his work. I refer to it here only in 
order to show that the opinion now under discus- 
sion is by no means confined to mathematicians, but 
has been recognized and adopted by men who cer- 
tainly cannot be charged with being subject to a ma- 
thematical bias. 

§ 5. Before proceeding to criticise this opinion 
I would make one remark upon it which has been 
constantly overlooked. It should be borne in mind 
that, even were this view of the subject not actually 
incorrect, it would nevertheless be insufficient for the 
purpose of a definition, inasmuch as variation of be- 
lief is not confined to Probability. It is a property 
with which that science is concerned, no doubt, but 
it is a property which meets us in many other di- 
rections as well. In every case in which we extend 
our inferences by Induction or Analogy, or depend 
upon the witness of others, or trust to our own me- 



SECT. 5.] LOGIC OF CHANCE. 63 

mory of the past, or come to a conclusion through 
conflicting arguments, or even make a long and com- 
plicated deduction by mathematics or logic, we have 
a result of which we can scarcely feel as certain as 
of the premises from which it was obtained. In all 
these cases then we are conscious of varying quan- 
tities of belief, but are the laws according to which 
the belief is produced and varied the same ? If they 
cannot be reduced to one harmonious scheme, if in 
fact they can be brought to nothing but a number 
of different schemes each with its own body of laws 
and rules, then it is in vain to endeavour to force 
them into one science. 

This opinion is strengthened by observing that 
most of the writers who adopt the definition in ques- 
tion do pi*actically dismiss from consideration most of 
the above-mentioned examples of diminution of belief, 
and confine their attention to classes of events which 
have the property discussed in Chap. I. viz. * ignorance 
of the few, knowledge of the many.' It is quite true 
that considerable violence has to be done to some of 
these examples, by introducing exceedingly arbitrary 
suppositions into them, before they can be forced to 
assume a suitable form. But still I have little doubt 
that, if we carefully examine the language employed, 
we shall find that in almost every case assumptions 
are made which virtually imply that our knowledge 
of the individual is derived from propositions given 



64 LOGIC OF CHANCE. [cHAP. HL 

in the typical form described in Chap. L This will 
be more folly proved when we come to consider some 
common misapplications of the science. 

§ 6. Even then, if the above-mentioned view of 
the subject were correct, it would yet be insufficient 
for the purpose of a definition ; but it is at least very 
doubtful whether it is correct. Before we could pro- 
perly assign to the belief side of the question the 
prominence given to it by Professor De Morgan and 
others, certainly before the science could be defined 
from that side, it would be necessary, I think, to 
establish the two following positions, against both of 
which strong objections can be brought. 

(1) That our belief of every proposition is a thing 
which we can, strictly speaking, be said to 
measure. There must be a certain amount 
of it in every case, which we can realize some- 
how in consciousness and refer to some 
standard so as to pronounce upon its value. 

(2) That the value thus apprehended is the cor- 
rect one according to the theory, viz. that 
it is the exact fraction of frill conviction that 
it should be. This statement will perhaps 
seem somewhat obscure at first; it will be 
explained presently. 

§ 7. (I) Now, in the first place, as regards the 
difficulty of obtaining any measure of the amount of 
our belief One source of this difficulty is too ob- 



SECT. 7.] LOGIC OP CHANCE. 65 

vioiis to have escaped notice ; this is the disturbing 
influence produced on the quantity of belief by any 
strong emotion or passion, A deep interest in the 
matter at stake, whether it excite hope or fear, plays 
great havoc with the belief-meter, so that we must 
assume the mind to be quite unimpassioned in weigh- 
ing the evidence. This is noticed and acknowledged 
by Laplace and others; but these writers seem to 
assume it to be the only source of error, and also to 
be of comparative unimportance. Even if it were 
the only source of error I cannot see that it would 
be unimportant. We experience hope or fear in so 
wery many instances, that to omit such influences 
fiom consideration would be almost equivalent to saying 
that whilst we profess to consider the whole quantity 
of our belief we will in reality consider only a por- 
tion of it. Very strong feelings are, of course, ex- 
ceptional, but we should nevertheless find that the 
emotional element^ in some form or other, makes it- 
self felt on almost every occasion. It is very seldom 
that we cannot speak of our surprise or expectation in 
reference to any particular event. Both of these ex- 
pressions, but especially the former, seem to point 
to something more than mere belief. I know that 
the word ^expectation' is generally defined in treatises 
on Probability as equivalent to belief; but I doubt 
i^hether any one who attends to the popular use of 
the terms would admit that they were exactly syno- 



66 IX)GIC OP CHANCE. [CHAP. IH. 

nymous. Be this however as it may, the emotional 
element is present upon almost every occasion, and 
its disturbing influence therefore is constantly at 
work. 

§ 8. Another cause, which co-operates with the 
former, is to be found in the extreme complexity and 
variety of the evidence on which our belief of any 
proposition depends. Hence it results that our belief 
is one of the most fugitive and variable things possible, 
so that we can scarcely ever get sufficiently clear hold 
of it to measure it. This is not confined to the times 
when our minds are in a turmoil of excitement 
through hope or fear. In our calmest moments we 
shall And it no easy thing to give a precise answer to 
the question, how firmly do I hold this or that belief? 
There may be one or two prominent arguments in 
its favour, and one or two coiTCsponding objections 
against it, but this is far fi*om comprising all the 
causes by which our state of belief is produced. Be- 
cause such reasons as these are all that can be prac- 
tically introduced into oral or written controversies, 
we must not conclude that it is by these only that 
our conviction is influenced. On the contrary, our 
conviction generally rests upon a sort of chaotic basis 
composed of an infinite number of inferences and 
analogies of every description, and these moreover dis- 
toi-ted by our state of feeling at the time, dimmed 
by the degree of our recollection of them afterwards. 



SECT. 8.] LOGIC OP CHANCE. 67 

and probably received from time to time with varying 
force according to the way in which they happen to 
combine at the moment. To borrow a striking illus- 
tration from Abraham Tucker, the substructure of 
oar convictions is not so much to be compared to the 
solid foundations of an ordinary building, as to the 
piles of the houses of Rotterdam which rest somehow 
in a deep bed of soft mud. They bear their weight 
securely enough, but it would not be easy to point 
out accurately the dependence of the different parts 
upon one another. Directly we begin to think of the 
amount of our belief, we have to think of the argu- 
ments by which it is produced — in fact, these argu- 
ments will intrude themselves without our choice. 
As each in turn flashes through the mind, it modifies 
the strength of our conviction ; we are like a person 
listening to the confused hubbub of a crowd, where 
there is always something arbitrary in the particular 
sound we choose to listen to. There may be reasons 
enough to suffice abundantly for our ultimate choice, 
but on examination we shall find that they are by no 
means apprehended with the same force at different 
times. The belief produced by some strong argu- 
ment may be very decisive at the moment, but jit 
will often begin to diminish when the argument 
is not actually before the mind. It is like being 
dazzled by a strong light ; the impression still re- 
mains, but begins almost immediately to fade away. 

5—2 



gg LOGIC OP CHANCE. [CHAP. IIL 

I tHnk tliat this is the case, however we try to limit 
the sources of our conviction. 

§ 9. (II) But supposing that it were possible to 
strike a sort of average of this fluctuating state, should 
we find this average to be of the amount assigned by 
theory ? In other words, is our natural belief in the 
happening of two difierent events in direct proportion 
to the frequency with which those events happen in 
the long runl There is a lottery with 100 tickets 
and ten prizes ; is a man's belief that he will get a 
prize fairly represented by one-tenth of certainty? 
The mere reference to a lottery should be sufficient to 
disprove this. Lotteries have flourished at all times, 
and have never failed to be abundantly supported, in 
spite of the most perfect conviction, on the part of 
many of those who put into them, that in the long 
run all will lose. Deductions should undoubtedly be 
made for those who act from superstitious motives, 
from belief in omens, dreams, <kc. But apart from 
these, and supposing any one to come fortified by 
all that mathematics can do for him, I cannot be- 
lieve that his natural impressions about single events 
would be always what they should be according to 
theory. Are there many who can honestly declare 
that they would have no desire to buy a single ticket ? 
They would probably say to themselves that the sum 
they paid away was nothing worth mentioning to 
lose, and that there was a chance of gaining a great 



SECT. 10.] LOGIC OF CHANCE. 69 

deal ; in other words, they are not apportioning their 
belief in the way that theory assigns. 

What beara out this view is, that the same per- 
sons who wonld act in this way in single instances, 
would often not think of doing so in any but single 
instances. In other words, the natural tendency is to 
attribute too great an amount of belief where it is 
or should be small; i. e. to disparage the risk in pro- 
portion to the contingent advantage. They would 
very likely, when argued with, attach disparaging 
epithets to this state of feeling by calling it an un- 
accountable fascination, or something of that kind, 
but of its existence there can be little doubt. I am 
speaking now of what is the natural tendency of our 
minds, not of that into which they may at length be 
disciplined by education and thought. If, however, 
educated persons have succeeded for the most part in 
controlling this tendency in games of chance, the 
* spirit of reckless speculation' has scarcely yet been 
banished from commerce. On examination, this tend- 
ency will be found, I think, so imiversal in all ages, 
ranks, and dispositions, that it would be inadmissible 
to neglect it in order to bring our supposed instincts 
more closely into accordance with the commonly re- 
ceived theories of Probability. 

§ 10. There is another aspect of this question 
which has been often overlooked, but which seems to 
deserve some attention. Granted that we have an 



70 LOGIC OF CHANCE. [CHAP. IIL 

instinct of credence, why should it be aasumed that 
it must be just of that intensity which subsequent 
experience will justify ? Our instincts are implanted 
in us by our Creator, and are intended to act immedi- 
ately and unconsciously. They are, however, subject 
to control, and have to be brought into accordance 
with what we believe to be true and right. In other 
departments of psychology we do not assume that 
every spontaneous prompting of nature is to be left 
just as we find it, or even that on the average^ omit- 
ting individual variations, it is set at that pitch that 
will be found in the end to be the best when we 
come to think about it and assign it its rules. Take, 
for example, the case of resentment. Here we have 
an instinctive tendency, and one that on the whole 
is good in its results. But moralists are agreed that 
almost all our efforts at self-control are to be directed 
towards subduing it and keeping it in its right direc- 
tion. It is assumed to be given as a sort of rough 
protection, and to be set, if one might so express 
oneself, at too high a pitch to be deliberately and 
consciously acted on in society. May not something 
of this kind be the case also with our belief 1 I only 
make a passing reference to this point here, as on 
the theory of Probability adopted in this work it does 
not seem to be at all material to the science. But 
it seems a strong argument against the expediency 
of conmiencing the study of the science from the sub- 



8EC3T. 10.] LOGIC OP CHAITCE. 71 

jectiye side, or even of assigning any great degree of 
prominence to this side. 

That men do not believe in exact accordance with 
this theory must have struck almost every one, but 
this has probably been considered as mere exception 
and irregularity ; the assumption being made that on 
the average, and in far the majority of cases, they 
(do so believe. As stated above, I think it very 
doubtful whether the tendency which has just been 
discussed is not so universal that it might with £gu: 
more propriety be called the law than the exception. 
And it may be better that it should be so : many 
good results may follow from that cheerful disposition 
which induces a man sometimes to go on trying after 
some great good, the chance of which he overvalues. 
He will keep on through trouble and disappointment, 
without serious harm perhaps, when the cool and 
calculating bystander sees plainly that his 'measure 
of belief is much higher than it should be. So, too, 
the tendency also so common, of underrating the 
chance of a great evil may also work for good. To 
many men death might be looked upon as an almost 
infinite evil, at least they would so regard it them- 
selves ; suppose they kept this contingency constantly 
before them at its right value, how would it .be 
possible to get through the practical work of lifel 
Men would be stopping indoors because if they went 
out they might be murdered or bitten by a mad dog. 



73 UKSC or CBASGB. [chap. m. 

I am not adroctting a l e lunfc to our inatmcts ; when 
we hare once rcftdied the critical and conacioiis state, 
it is not possible to do so; but it shoold be noticed 
tbat the adrantage gained bj eocreciing ihem is at 
best bat a balanced one. What is most to onr pre- I 

sent purpose, it snggests the inexpediency of attempt- 
ing to fonnd an exact theoiy on what may afterwards 
prove to be a mere instinct^ muuithoriaed in its full 
extent by experience. 

§ 11. It may be relied, that thoo^ people, as 
a matter of &ct^ do not apportion belief in this exact i 

way, yet they aught to do so. The purport of this re- 
mark will be examined presently ; I will only say here 
that it grants all that I am contending for. For it 
admits that the d^ree of our belief is capable of 
modification, and may need it. But in accordance | 

with what is the belief to be modified) obyionsly 
in accordance with experience; it cannot be trusted 
to by itself, bat the firaction at which it is to be rated 
most be determined by the comparatiye frequency of 
the events to which it refers. Experience^ then, for- j 

nishing the standard, it is surely most reasonable to ! 

start from this experience, and to found the theory of 
our process upon it. i 

If we do not do this it should be observed that 
we are detaching Probability altogether from the 
study of things external to us, and making it nothing 
else in effect than a portion of Psychology. If wo 



.SBCT. 12.] LOGIC OF CHANCE. 73 

refuse to be controlled by experience, but confine our 
attention to the laws according to which belief is 
naturally or instinctively compounded and distributed 
in our minds, we have no right then to appeal to 
experience afterwards even for illustrations, unless 
under the express understanding that we do not 
guarantee its accuracy. Our belief in some single 
events, for example, might be correct, and yet that 
in a compound of several (if derived merely from our 
instinctive laws of belief) very possibly might not 
be correct, but might lead us to error if we deter- 
mined to act upon it. Even if the two were in ac- 
cordance, this accordance would have to be proved, 
which would lead us round, by what I cannot but 
think a circuitous process, to the point which has 
been already chosen for commencing with. 

§ 12. Professor De Morgan seems to imply that 
the doctrine criticized above finds its justification 
from the analogy of Formal Logic. I confess I can- 
not see much force in the analogy. Formal Logic 
is based upon the assumption that there are laws of 
mind as distinguished from laws of things, and that 
these laws of mind can be ascertained and studied 
without taking into account their reference to any 
particular object. But to support this assumption a 
postulate has to be claimed, or else a consequence 
&ced. The postulate is, that the laws of the things 
are so fiir in harmony with those of our minds that 



74 LOGIC OF CHANCE. [cHAP. in. 

we may be certain that any exercise of oxir minds 
will not lead ns into contradictions in practice. If 
this postulate be not granted we must then be pre- 
pared to brave any consequences that may follow from 
a want of such harmony. It is supposable, as some 
logicians seem ready to admit, that the laws of mat- 
ter should be defiantly at variance with those of 
mind. So much the worse for us, but we cannot 
help it; we must go on thinking in accordance with 
. our laws, for they are unhappily fixed for ever and 
invariable, and we must be content to take the con- 
sequences. But, as was briefly stated in § 11, no 
such distinction can be drawn in the case of laws 
of belief as we find them in Probability. Our in- 
stincts of credence are unquestionably in frequent 
hostility with experience; and what do we do. then? 
We simply modify the instincts into accordance with 
the things. We are constantly performing this prac- 
tice, and no cultivated mind would find it possible 
to do anything else. No man would think of di- 
vorcing his belief from the things on which it was 
exercised, or of thinking that the former had any- 
thing else to do than to follow the lead of the latter. 
Whatever then may be the claims of Formal Logic 
to rank as a sepamte science, it cannot, I think, 
furnish any support to the theory of Probability as 
conceived by some mathematicians. 

§ 13. I have examined the doctrine in question 



SECT. 14.] LOGIC OF CHANCE. 75 

with a minuteness T^hich may seem tedious, but in 
consequence of the eminence of its supporters it would 
liave been presumptuous to have rejected it without 
the strongest grounds. The objections which have 
just been urged might be summarized as follows ; — 
the amount of our belief of any given proposition, 
supposing it to be in its nature capable of determina- 
tion (which is extremely doubtful), depends upon a 
great variety of causes, of which statistical frequency, 
— ^the subject of Probability — ^is but one. That even 
if we confine our attention to this ooe cause, the 
natural amount of our belief is not necessarily what 
theory would assign, but has to be checked by ap- 
peal to experience. The subjective side of Pro- 
bability therefore, though very interesting and well 
deserving of examination, seems a mere appendage of 
the objective, and affords in itself no safe ground for 
a science of inference. 

§ 14. The conception then of the science of Pro- 
bability as a science of the laws of belief seems to 
me to break down at every point. We must not 
however rest content with such merely negative 
criticism. The degree of belief we entertain of a 
proposition may be hard to get at accurately, and 
when obtained may be often wrong, and need there- 
fore to be checked by an appeal to the objects of 
belief Still in popular estimation we do seem to 
be able with more or less accuracy to form a gra- 



76 LOGIC OF CHANCE. [cHAP. ni.' 

duated scale of intensity of belie£ What we have 
to examine now is whether this be possible, and, if 
so, what is the explanation of the fact ] 

That it is generally believed that we can form 
snch a scale scarcely admits of doubt. There is a 
whole vocabulary of common expressions such as,' I 
feel almost sure; I do not feel quite certain; I am 
less confident of this than of that, <fec. When we 
make use of any one of these phrases we never doubt 
that we have a distinct meaning to convey by means 
of it. Nor do we feel much at a loss, under any 
given circumstances, as to which of these expressions 
we should employ in preference to the others. If we 
were asked to arrange in order, according to the 
intensity of the belief with which we respectively 
hold them, things broadly marked off from one an- 
other, we could do it from our consciousness of be- 
lief alone, without a fresh appeal to the evidence 
upon which the belief depended. Passing over the 
looser propositions which are used in common con- 
versation, let us take but one simple example from 
amongst those which furnish numerical data. Do I 
not feel more certain that some one will die this week 
in the town, than in the street in which I live 1 and 
if the town contain a population one hundred times 
greater than that in the street, would not almost taiy 
one assert unhesitatingly that he felt a hundred times 
more sure of the first proposition than of the second I 



1 



SECT. 15.] LOaiC OP CHANCE. 7T 

Here then a problem proposes itself. If popular 
opinion, as illustrated in common language, be cor- 
rect, — and very considerable weight must of course 
be attributed to it, — there does exist something which 
we call partial belief in reference to any proposition 
of the numerical kind described above. Now what 
we want to do is to find some test or justification of 
this belief, to obtain in fact some intelligible answer 
to the question, Is it correct? We shall find in- 
cidentally that the answer to this question will throw 
a good deal of light upon another question, viz. what 
is the meaning of this partial belief? 

§ 15. We shall find it advisable to commence by 
ascertaining how such enquiries as the above would 
be answered in the case of ordinary full belief. Such 
a step will not offer the slightest difficulty. Suppose, 
to take a simple example, that we have obtained the 
following proposition, — whether by Induction, or the 
rules of ordinary Logic, does not matter for our pre- 
sent purpose, — ^that a mixture of oxygen and hydrogen 
is explosive. Here we have an inference, and conse- 
quent belief of a proposition. Now suppose there were 
any enquiry as to whether our belief were correct, 
what should we do ? The simplest way of settling the 
matter would be to find out by a distinct appeal to 
experience whether the proposition was true. Since 
we are reasoning about things, the justification of the 
belief, that is, the test of its correctness, would be 



78 LOGIC OP CHANCE. [CHAP. m. 

most readily found in the truth of the proposition! 
If by any process of inference I have come to believe 
that a certain mixture will explode, I consider my 
belief to be justified, that is to be correct, if under 
proper circumstances the explosion always occurs; 
if it does not occur the belief was wrong. 

Such an answer, no doubt, goes but a little way, 
but it is sufficient for our present purpose. In all 
inferences about things, in which the wmowrd of our 
belief is not taken into account, such an explanation 
as the above is quite sufficient; it woidd be the 
ordinary one in any question of science. It is more- 
over perfectly intelligible, whether the conclusion is 
particular or universal Whether we believe that 
some men die, or that all men die, our belief may 
with equal ease be justified by the appropriate traiii 
of experienca 

§ 16. But when we attempt to apply the same 
test to 'partial belief, we shall find ourselves reduced 
to an awkward perplexity. A difficulty now emerges 
which has been singularly overlooked by those who 
have treated of the subject. As a simple example 
will serv« our purpose, we will take the case of a 
penny. I am about to toss one up, and I therefore 
half-believe, to adopt the current language, that it 
will give head. Now it seems to be overlooked that 
if we appeal to the event, as we did in the case last 
examined, our belief must inevitably be wrong, and 



SECT. 16.] LOGIC OF CHANCE. 79 

therefore the test above mentioned will fail. For the 
thing must either happen or not happen ; i. e. in this 
case the penny must either give head, or not give 
it ; there is no third alternative. But whichever way 
it occurs, our half-belief, so far as such a state of 
mind admits of interpretation, must be wrong. If 
head does come, I am wrong in not having expected 
it enough ; for I only half believed in its occurrence. 
If it does not happen, I am equally wrong in having 
expected it too much; for I half believed in its 
occurrence, when in fact it did not occur at all. 

The same difficulty will occur in every case in 
which we attempt to justify our state of partial be- 
lief in a single contingent event. Let us take another 
example, slightly differing from the last. A man 
is to receive £1 if a die gives six, to pay Is. if it 
gives any other number. It will generally be ad- 
mitted that he ought to give 28. 6d, for the chance, 
and that if he does so he will be paying a fair sum. 
This example only differs from the last in the fact 
that instead of simple belief in a proposition, we have 
taken what mathematicians call *the valiie of the 
expectation.' In other words, we have brought into 
greater prominence, not merely the belief, but the 
conduct which is founded upon the belief. But pre- 
cisely the same difficulty recurs here. For appealing 
to the event, — ^the single event that is,^ — ^we see that 
one or other party must lose his money without 



80 LOGIC OP CHANCE. [CHAP. UI. 

compensation. In wliat sense tlien can such an ex- 
pectation be said to be a fair one 1 

§ 17. A possible answer to this, and so far as 
I can see the only possible answer, will be, that what 
we really mean by saying that we half believe in 
the occurrence of head is to express our conviction 
that head will certainly happen on the average every 
other time. And similarly, in the second example, 
by calling the sum a fair one it is meant that iu 
the long run neither party will gain or lose. I am 
not sure that such an answer will be made ; if it is 
I almost entirely agree with it. As we shall recur 
to it presently, the only notice that can be taken 
of it at this point is to call attention to the fact 
that it entirely abandons the whole question in dis- 
pute, for it admits that this partial belief does not in 
any strict sense apply to the individual event, for it 
clearly cannot be justified there. At such a result 
indeed we cannot be surprised, at least we cannot on 
the theory adopted throughout this Essay. For bear- 
ing in mind that the employment of Probability 
postulates ignorance of the single event> it is not easy 
to see how we are to justify any other opinion or 
statement about the single event than a confession of 
such ignorance. 

§ 18. So far then we do not seem to have made 
the slightest approximation to a solution of the par> 
ticular question " now under examination. The more 



SECT. 18.] LOGIC OF CHANCE. 81 

closely we have analysed special examples, the more 
immistakeably are we brought to the conclusion that 
in the individual instance no justification of anything 
like quantitative belief is to be found ; at least none 
is to be found in the same sense in which we expect 
it in ordinary scientific conclusions, whether Inductive 
or Deductive. And yet we have to face and account 
for the fact that common impressions, as attested by 
a whole vocabulary of common phrases, are in favour 
of the existence of this quantitative belief. How are 
we to account for this ? If we appeal to an example 
again, and analyze it somewhat more closely, we may 
yet find our way to some satisfactory explanation. I 
offer it however with some diffidence, for though it is 
the best to which, after much reflection, I can see 
my way, the enquiry is one which more properly 
belongs to those who have made a longer and more 
profound study of Psychology than I have been able 
to do. 

In our previous analysis we found it sufficient to 
stop at an early stage, and to give as the justification 
of our belief the fact of the proposition being true. 
Stopping however at that stage we have found this 
explanation fiedl altogether to give a justification of 
partial belief; fail, that is, when applied to the in- 
dividual instance. Suppose then we advance a step 
farther in the analysis, and ask again what is meant 
by the proposition being true ? This introduces us, 

6 



82 LOGIC OF CHANCE. [CHAP. III. 

of course, to a very long and intricate path, but in 
the short distance along it which we shall advance, 
we shall not I hope find any very serious difficulty. 
As before, we will illustrate the analysis by first 
applying it to ordinary full belief. 

§ 19. Whatever opinion then may be held about 
the essential nature of belief, it will probably be ad- 
mitted that a readiness to act upon the proposition 
believed is an inseparable accompaniment of that 
state of mind. There can be no alteration in the 
belief without a possible alteration in the conduct, 
nor anything in the conduct which is not connected 
with something in the belief. We will first take an 
example in connection with the penny, in which, as 
I have said, there is full belief; we will analyse it 
a step farther than we did before, and then attempt 
to apply the same analysis to an example of a similar 
kind, but one in which the belief is partial instead 
of full. 

§ 20. Suppose that I am about to throw a penny 
up, and contemplate the prospect of its falling upon 
one of its sides and not upon its edge. Whatever 
else may be implied in our belief we certainly mean 
this; that we are ready to stake our conduct upon 
its falling thus. All our betting, and everything 
else that we do, is carried on upon this supposition. 
Any risk whatever that might ensue upon its falling 
otherwise will be incurred without fear. This, it must 



] 



SECT. 20.] LOGIC OP CHANCE. 83 

be observed, is equally the case whether we are speak- 
ing of a single throw or of a long succession of throws. 

But now let us take the case of a penny falling, 
not upon one side or the other, but upon a given side, 
head. To a certain extent this example resembles 
the last. "We are perfectly ready to stake our 
conduct upon what comes to pass in the long run. 
When we are considering the result of a large num- 
ber of throws, we are ready to act upon the supposi- 
tion that head comes every other time. If, e. g. we 
are betting upon it, we shall not object to paying £1 
every time that head comes, on condition of receiving 
£1 every time that head does not come. This is 
nothing else than the trcmslation, as we may call it, 
into practice, of our belief that head and tail i:>ccur 
equally often. 

Now it will be ob^ous, on a moment's consideration, 
that our conduct is capable of being slightly varied, of 
being vaiied, I mean, in form, whilst it remains iden- 
tical in result. It is clear that to pay £1 every time we 
lose, and to get £1 every time we gain, comes to pre- 
cisely the same thing, in the case under consideration, 
as to pay ten shillings every time without exception, 
and to receive £1 every time that head occurs. It is 
so, because heads occur, on the average, every other 
time. In the long run the two results coincide, but 
there is a marked difference between the two cases, 
considered individually. The difference is two-fold. 



84 LOGIC OP CHANCE. [CHAP. III. 

In the first place we have slid from the notion of a 
payment every other time, and come to that of one 
made every time. In the second place, what we pay 
every time is half of what we get in the cases in 
which we do get anything. The difference may seem 
slight; but mark the effect when our conduct is 
translated back again into the subjective condition 
upon which it depends, viz. into our belief It is 
in consequence of such a translation, as it appears to 
me, that the notion has been acquired that we have 
an accurately determinable amount of belief as to 
every such proposition. To have losses and gains of 
equal amount, and to incur them equally often, was 
the experience connected with our belief that the 
two events, head and tail, would occur equally often. 
This was quite intelligible, for it referred to the long 
run. To find that this could be commuted for a 
payment made every time without exception, a pay- 
ment, observe, of half the amount of what we occa- 
sionally receive, has very naturally been interpreted 
to mean that there must be a state of half-belief 
which refers to each individual throw. 

§ 21. One such example, of course, does not 
go far towards establishing a theory. But the reader 
will bear in mind that almost all our conduct tends to- 
wards the samo result, — ^that it is not in betting only, 
but in every circumstance in which we have to count 
the events, that such a numerical apportionment of 



SECT. 21.] LOGIC OF CHANCE. 85 

our conduct is possible. Hence, by the ordinary 
principles of association, it would appear exceed- 
ingly likely that, not exactly a numerical condition 
of mind, but rather, numerical associations become 
inseparably connected with each particular event. 
Once in six times a die gives ace; a knowledge of 
this fact, taken in combination with all the practical 
results to which it leads, produces, one cannot doubt, 
an inseparable notion of one-sixth connected with 
each single throw. But it surely cannot be called 
belief to the amount of one-sixth ; at least it admits 
neither of justification nor explanation in these single 
cases, to which alone the fractional belief, if such 
existed, ought to apply. 

It is in consequence, I apprehend, of such associa- 
tion that we act in such an unhesitating manner in 
reference to any single contingent event, even when 
we have no expectation of its being repeated. A die 
is going to be thrown up once, and once only. I bet 
5 to 1 against ace, not, as is commonly asserted, 
because I feel one-sixth part of certainty in the occur- 
rence of ace ; but because I know that such conduct 
would be justified in the long run of such cases, and 
I apply to the solitary individual the same rule that 
I should apply to it if I knew it were one of a long 
series. This accounts for my conduct being the same 
in the two cases; by association, moreover, we probably 
experience very similar feelings in regard to them botL 



86 lOGIC OF CHANCE. [CHAP. HL 

§ 22. And here^ on my view of the subject, we 
might st<^ We are bonnd to explain the ' measure of 
our belief' in the occurrence of a single event when 
we judge solely from the statistical frequency with 
which sudi events occuTy for such a series of events 
was our starting-point; but we are not bound to 
inquire whether in every case in which persons have, 
or claim to have, a certain measure of belief there 
must be such a series to which to refer it, and by 
which to justify it. Those who start from the sub> 
jective dde, and r^ard Probability as the ^ience of 
quantitative belief, are obliged to do this, but we are 
free from the obligation. 

Still the question is one which is so naturally 
raised in connection with this subject^ that it cannot 
Ije altogether passed by. I think that to a consider- 
able extent such a justification as that mentioned 
above will be found applicable in other cases. T^e 
fact is that we are very seldom called upon to decide 
and act upon a single contingency which cannot be 
viewed as being one of a series. Experience intro* 
duces us, it must be remembered, not merely to a 
succession of events neatly arranged in a single series 
(as we have hitherto assumed them to be for the pur- 
l>ose of illustration), but to an infinite number belong- 
ing to a vast variety of different series. A man is 
obliged to be acting, and therefore exercising his 
belief about one thing or another almost the whole of 



SECT. 22.] LOGIC OF CHANCE. 87 

every day* of Ms life. Any one person will have to 
decide in his time about a multitude of events, each 
one of which will never recur again within his own 
experience. But by the very fact of there being a 
multitude, though they are all of different kinds, we 
shall still find that order is maintained, and so a 
course of condtiot can be justified. In a plantation of 
trees we should find that there is order of a certain 
kind if we measure them in any one direction, the 
trees being on an average about the same distance 
from each other. But a similar order would be 
found if we were to examine them in any other direc- 
tion whatsoever. So in nature generally; there is 
regularity in a succession of events of the same kind. 
But there may also be regularity if we form a series 
by taking successively a number out of totally distinct 
kinds. 

It is in this circumstance that we find an exten- 
sion of the practical justification of the measure of our 
belief A man, say, buys a life annuity, insures his 
life on a railway, puts into a lottery, and so on. Now 
we may make a series out of these acts of his, though 
each is in itself a single event which he never intends 
to repeat. His conduct, and therefore his belief, 
measured by the result in each individual instance, 
will not be justified, but the reverse, as shewn in § 16. 
Could he indeed repeat each kind of action often 
enough it would be justified, but from this, by the 



88 LOGIC OF CHANCE. [CHAP. III. 

conditions of life, he is debarred. Kow it is perfectly 
conceivable that in the new series, formed by his sac> 
eessive acts of different kinds, there should be no re- 
gularity. As a matter of fact> however, it is found 
that there is regularity. In this way the equalization 
of his gains and losses for which he cannot hope in 
annuities, insurances, and lotteries separately, may- 
yet be secured to him out of these events taken col- 
lectively. If in each case he values his chance at its 
right proportion (believing accordingly) he will in the 
course of his life neither gain nor lose. And in the 
same way if, whenever he has the alternative of differ- 
ent courses of conduct, he acts in accordance with the 
estimate of his belief described above, i.e. chooses the 
event whose chance is the best, he will in the end 
gain more in this way than by any other course. By 
the existence, therefore, of these cross-series, as we 
may term them, there is an immense addition to the 
number of actions which may be &irly considered to 
belong to those courses of conduct which offer many 
successive opportunities of equalizing gains and losses. 
All these cases then may be regarded as admittiDg of 
justification in the way now under discussion. 

§ 23. In the above remarks it will be observed 
that we have been giving what is to be regarded as a 
justification of his belief from the point of view of the 
individual agent himself. If we suppose the existence 
of an enlarged fellow-feeling, the applicability of such 



SECT. 23.] LOGIC OF CHANCE. 89 

a justification becomes still more extensive. We can 
assign a very intelligible sense to the assertion tliat it 
is 999 to 1 that I shall not get a prize in a lottery, 
even if this be stated in the form that my belief in 
my so doing is represented by the fraction tuW*^ of 
certainty. Properly it means that in a very large 
number of throws I should gain once in 1000 times. 
If we admit other contingencies of the same kind, as 
described in the last section, each individual may be 
supposed to reach to something like this experience 
within the limits of his own life. He could not do it 
in this particular line of conduct alone, but he could 
do it in this line combined with others. Now intro- 
duce the possibility of each man feeling that the gain 
of others offers some analogy to his own gains, which 
we may conceive his doing except in the case of the 
gains of those against whom he is directly competing, 
and the above justification becomes still more exten- 
sively applicable. 

The following, I think, would be a feir illustration 
to test this view. I know that I must die on some day 
of the week, and there are but seven days. My belief, 
therefore, that I shall die on a Sunday is one-seventh. 
Here the contingent event is clearly one that does 
not admit of repetition; and yet would not the belief 
of every man have the value assigned it by the for- 
mula 1 I think that the same principle will be found 
to be at work here as in the former examples. It is 



90 LOGIC OF CHANCE. [cHAP. HI. 

quite true that I have only the opportunity of dying 
once myself, but I am a member of a class in which 
deaths occur with frequency, and I form my opinion 
upon evidence drawn from that class. I^ for example, 
I had insured my life for £1000, I should probably 
demand £7000 in case the office declared that it 
would only pay in the event of my dying on a Sun- 
day. / might not find the arrangement an equitable 
one, but mankind at large, in case they acted on such 
a principle, might fairly commute their aggregate 
gains in such a way. 

§ 24. The results of the last few sections might 
be summarized as follows : — ^the different amounts of 
belief which we entertain upon different events, and 
which are recognized by various phrases in common 
use, have undoubtedly some meaning. But their 
meaning, and certainly their justification, is to be 
sought in the series of corresponding events to which 
they belong; in regard to which it may be shown that 
far more events are capable of being referred to a 
series than might be supposed at first sight. The test 
and justification of belief are to be found in conduct; 
in this test applied to the series as a whole, there 
IS nothing peculiar, it is like acting on our belief 
about any single thing. But so applied, it is applied 
successively to each of the individuals of the series; . 
here our conditct generally admits of being in some 
way divided up numerically (I see no better way of 



SECT. 25.] LOGIC OF CHANCE. 91 

describing it) in reference to each particular event; 
and this has been understood to denote a certain 
amount of belief which should be a fraction of cer- 
tainty. Probably on the principles of association, a 
pecidiar condition of mind is produced in reference to 
each single event. And these associations are not 
unnaturally retained even when we contemplate any 
one of these single events isolated from any series to 
which it belongs. When it is found alone we treat it, 
and feel towards it, as we do when it is in company 
with the rest of the series. 

§ 25. We may now see, more clearly than we 
could before, why it is that we are free from any 
necessity of assuming the existence of causation, in 
the sense of necessary invariable sequence, in the case 
of the events which compose our series. Against 
such a view it is sometimes urged, that we constantly 
talk of the probability of a single event; but how 
can this be done, it would probably be said, if we 
once admit the possibility of that event occurring 
fortuitously ? Take an instance from human life ; the 
average duration of the lives of any batch of men 
aged thirty will be about thirty-four years. We say 
therefore to any individual of them. Your expect- 
ation of life is twenty-five years. But how can this 
be said if we admit that his life is liable to be des- 
titute of all I'egular sequence of cause and effect? 
I reply that the denial of causation enables us to 



92 LOGIC OP CHANCE. [CHAP. III. 

say neither more nor less than its assertion, in refer- 
ence to the individual life, for of this we are igno- 
rant in each case alike. £7 assigning, as above, an 
expectation in reference to the individual, we msaai 
nothing more than to make a statement about the 
average of his class. Whether there be causation or 
not in these individual cases does not affect our know- 
ledge of the average, for this by supposition rests on 
independent experience. The legitimate inferences 
are the same in each case, and of equal value. The 
only difference is that in the one case we have forced 
upon our attention the impropriety of talking of the 
'proper' expectation of the individual, owing to the 
fact that all knowledge of its amount is formally 
impossible ; in the other case the impropriety is over- 
looked from the fact of such knowledge being only 
practically unattainable. As a matter of fact the 
amount of our knowledge is the same in each case ; 
it is a knowledge of the average, and of that only.* 
§ 26. I conclude then that the limits within 
which we are thus able to justify the amount of our 
belief are far more extensive than might appear at 
first sight. Whether every case in which persons 
feel an amoimt of belief short of perfect confidence 
could be forced into the province of Probabiliiy is 
a wider question. Even, however, if the belief could 
be supposed capable of justification on its principles^ 
* For ft fuller discnflsioii of this, see Chap. ziv. 



i 



SECT. 27.] LOGIC OF CHANCE. 93 

its rules could never in such cases be made use of. 
Suppose, for example, that a father were in doubt 
whether to give a certain medicine to his sick chilcj. 
On the one hand the doctor declared the child would 
die unless the medicine were given; on the other, 
through a mistake, the father cannot feel quite sure 
that the medicine by him is the right one. It is con- 
ceivable that Laplace, in his conviction that every- 
thing is a probability, would declare that the man's 
belief had some 'value' (if he could only find out 
what it is), say nine-tenths; that this means that in 
nine cases out of ten in which he entertained a be- 
lief of that particular value it turned out that he 
was right. So with his doubt on the other side. 
Putting the two together there is but one course 
which, as a prudent man and a good father, he can 
possibly follow. It may be so, but when (as here) 
the identification of an event in a series depends 
OD. purely subjective conditions, as in this case upon 
the degree of vividness of his conviction, of which 
no one else can judge, no test is possible, and there- 
fore no proof can be found. One dare not dogmatise 
by denying, and therefore, when any person gets the 
start by an assertion, he must be left in the field. 
A question very closely connected with this will bo 
treated of in a future chapter on Common Misap- 
plications of the Theory of Probability. 

§ 27. So much then for the attempts, d6 fre- 



94 LOGIC OP CHANCB. [CHAP. HI. 

quently made, to found the science on a subjective 
basis ; they can lead, as I have endeavoured to show, 
to no satisfactory result. Still our belief is so in- 
separably connected with our action that something 
of a defence can be made for the attempts described 
above; but when it is attempted, as is often the 
case, to import other sentiments besides pure be- 
lief, and to find a justification for them also in the 
results of our science, the confusion becomes far 
worse. The following extract from Archbishop 
Thomson's Laws of Thought will show what kind of 
applications of the science are contemplated here : 
"In applying the doctrine of chances to that sub- 
ject in connexion with which it was invented — 
games of chance, — the principles of what has been 
happily termed * moral arithmetic' must not be for- 
gotten. Not only would it be difficult for a gamester 
to find an antagonist on terms, as to fortune and 
needs, precisely equal, but also it is impossible that 
with such an equality the advantage of a consider- 
able gain should balance the harm of a serious loss." 
" If two men," says Buffon, " were to determine to 
play for their whole property, what would be the 
effect of this agreement ? The one would only double 
his fortune, and the other reduce his to naught. 
What proportion is there between the loss and the 
gain 1 The same that there is between all and no- 
thing. The gain of the one is but a moderate sum, — 



SECT. 27.] LOGIC OP CHANGE. 95 

the loss of the other is numerically infinite, and 
morally so great that the labour of his whole life 
may not perhaps suffice to restore his property." 

As moral adyice this is all very true and good. 
But if it be regarded as a contribution to the sci- 
ence of the subject it is quite inappropriate, and 
seems calculated to cause confusion. The doctrine of 
chances pronounces upon certain kinds of events in 
respect of number and magnitude ; it has absolutely 
nothing to do with any particular person's feelings 
about these relations. We might as well append 
a corollary to the rules of arithmetic, to point out 
that although it is very true that twice two are four 
it does not follow that four sheep will give twice as 
much pleasure to the owner as two will. If two 
men play on equal terms their chances are equal; 
in other words, if they were often to play in this 
manner each would lose as frequently as he would 
gain. That is all that Probability can say; what 
under the circumstances may be the determination 
and opinions of the men in question, it is for them 
and them alone to decide. There are many persons 
who cannot bear mediocrity of any kind, and to whom 
the prospect of doubling their fortune would outweigh 
a greater chance of losing it altogether. They alone 
are the judges. 

If we will introduce such a balance of pleasure 
and pain the individual must make the calculation for 



96 LOGIC OF CHANCE. [CHAP. HI. 

himself. The supposition is that total ruin is very 
painful, partial loss painful in a less proportion than 
that assigned by the ratio of the losses themselves; 
the inference is therefore drawn that on the average 
more pain is caused by occasional great losses than by 
frequent small ones, though the -money value of the 
losses in the long run may be the same in each case. 
But if we suppose a country where the desire of 
spending largely is very strong and where from abund- 
ant production loss is easily replaced, the calculation 
might incline the other way. Under such circum- 
stances it is quite possible that more happiness might 
result from playing for high than for low stakes. The 
fact is that all emotional considerations of this kind 
are irrelevant; they are, at most, mere applications of 
the theory, and such as each individual is alone com- 
petent to make for himself. 

§ 28. It is by the introduction of such consider- 
ations as these that the really very simple and in- 
telligible Petersburg Problem has been so perplexed. 
Having already given some description of this pro- 
blem I will refer to it very briefly here. It presents 
us with a sequence of sets of throws for each of 
which sets I am to receive something. My receipts 
increase in proportion to the rarity of each particular 
kind of set, and each kind is found to grow more 
rare in a certain definite but unlimited order. By 
the wording of the problem, properly interpreted, I 



SECT. 28.] THE LOGIC OF CHAXCE. 97 

am supposed never to stop. Clearly therefore, how-* 
ever large a fee I pay for each of these sets, I shall 
be sure to make it up in time. The mathematical 
expression of this is, that I ought always to pay an 
infinite sum. To this the objection is opposed, that no 
sensible man would think of advancing even a large 
finite sum, say £1000. Certainly he would not; but 
why ] Because neither he nor those who are to pay 
him would be likely to live long enough for him to 
obtain throws good enough to remunerate him for one- 
tenth of his outlay; to say nothing of his trouble and 
loss of time. We must not suppose that the problem, 
as stated in the ideal form, will coincide with the 
practical form in which it presents itself in life. A 
carpenter might as well object to Euclid's second pos- 
tulate, because his plane came to a stop in six f^et 
on the plank on which he was at work. Many per- 
sons have fiiiled to perceive this, and have assumed 
that, besides enabling us to draw numerical inferences 
about the members of a series, the theory ought also 
to be called upon to justify all the opinions which 
average respectable men might be inclined to form 
about them, as well as the conduct they might choose 
to pursue in consequence. It is obvious that to enter 
upon such considerations as these is to diverge from 
our proper ground. We are concerned, in these cases, 
with the actions of men only, as given in statistics ; 
with the emotions they experience in the performance 

7 



98 THE LOGIC OF CHANCE. [CHAP. m. 

of these actions we have no direct concern whatever. 
The error is the same as if anj one were to oonfonnd, 
in political economy, value in use with value in ex- 
change, and ohject to measuring the value of a loaf by 
its cost of production, because bread is worth more to a 
man when he is hungry than it is just after his dinner. 

§ 29. One class of emotions indeed ought to be 
^cepted, which, from the apparent uniformiiy and con- 
sistency with which they show themselves in different 
persons and at different times, do really present some 
better claim to consideration. In connection with a 
science of inference they can never indeed be regarded 
as more than an accident of what is essential to the 
subject) but compared with other emotions they seem 
to be inseparable accidents. 

The reader will remember that attention was 
drawn in the earlier part of this chapter to the com* 
pound nature of the state of mind which we term 
belief. It is partly intellectual, partly also emotional; 
it professes to rest upon experience, but in reality the 
experience acts through the distorting media of hopes 
and fears and other disturbing agencies. So long as 
we confine our attention to the state of mind of the 
person who believes, it appears to me that these two 
parts of belief are quite inseparable. Indeed, to speak 
of them as two parts may convey a wrong impression; 
for though they spring from different sourceEi, they so 
entirely merge in one result as to produce what might 



n 



SECT. 29.] THE LOGIC OF CHANCE. 99 

be called a simple compound. Every kind of infer- 
ence, whether in probability or not, is liable to be 
disturbed in this way. A timid man may honestly 
believe that he will be wounded in a coming battle, 
when others, with the same experience but calmer 
judgmentH, see that the chance is too small to deserve 
consideration* But such a man's belief, if we look 
only to that) will not differ from sound belief. His 
conduct also in consequence of his belief will by itself 
afford no ground of discrimination; he will make his 
will as sincerely as a man on his death-bed. The only 
resource is to check his belief by appealing to past 
and current experience. This was advanced as an 
objection to the theory on which probability is re- 
garded as concerned primarily with laws of beliefl 
But on the view taken in this Essay in which we are 
supposed to be concerned with laws of inference about 
things, error and difficulty from this source vanish. 
Iiet us bear clearly in mind that we are concerned 
with inferences about things, and are always to test 
our belief by experience of the things, and whatever 
there may be in belief which does not depend on 
experience will disappear from notice. It is con- 
ceivable indeed that men might be in such a state 
of abject panic that their senses, at the time or after- 
wai*ds, were disturbed like their judgment beforehand. 
If so, we must appeal to the wider experience of other 
men of calmer mind, or to their own judgment in 

7—2 



100 THE LOGIC OP CHANCE. [cHAP. Ill, 

their better moments. . These are the ultimate, and 
apparently the only ultimate courts of appeal. 

§ 30. The only notice then that these emotions 
can claim as an integral portion of any science of 
inference is, that they should be rigidly excluded fix)m 
it. But if any of them are uniform and regular in 
their production and magnitude, they may be fairly 
admitted as accidental and extraneous accompani- 
ments. This is really the case to some extent with 
our surprise. This emotion does show a considerable 
degree of uniformity. The rarer any event is the 
more am I, in common with most other men, sur- 
prised at it when it does happen. This surprise 
may range through all degrees, from the most languid 
form of interest up to the condition which we term 
' being startled.' And since the surprise seems to be 
pretty much the same, under similar circimistances, at 
different times, and in the case of different persons, 
it is free from that extreme irregularity which is 
found in most of the other mental conditions which 
accompany the contemplation of unexpected events. 
Hence our surprise, though, as stated above, having 
no proper claim to admission into the science of Pro- 
bability, is such a constant and regular accompaniment 
of that which Probability is concerned with, that 
notice must often be taken of it Beferences will oc- 
casionally be found to this aspect of the question in 
the following chapters. 



SECT. 31.] THE LOGIC OF CHANCE. 101 

It may be remarked in passings for the sake of 
further illustration of the subject^ that this emotional 
accompaniment of surprise, to which we are thus able 
to assign something like a fractional value, differs 
in two important respects from the commonly accepted 
fraction of beliefl In the first place, it has what may 
be termed an independent existence ; it is intelligible 
by itsel£ The belief, as I endeavoured to show, 
needs explanation and finds it in our consequent 
conduct. Not so with the emotion ; this stands upon 
its own footing, and may be examined in and by 
itself Hence, in the second place, it is as applicable, 
and as capable of any kind of justification, in relation 
to the single event, as to a series of events. In this 
respect, as will be remembered, it offers a complete 
contrast to our state of belief about any one con- 
tingent event. May not these considerations help to 
account for the general acceptance of the doctrine, 
that we have a certain definite and measurable 
amount of belief about these events % I cannot help 
thinking that what is so obviously true of the emo- 
tional portion of the belief, has been unconsciously 
transferred to the other or intellectual portion of the 
compound condition, to which it is not applicable, and 
where it cannot find a justification. 

§ 31. A further illustration may now be given 
of the subjective view of Probability at present under 
discussion* 



102 THE LOGIC OP CHANCE. [CHAP. HI. 

An appeal to common language is always of ser- 
Tice, as the employment of any distinct word is 
generally a proof that mankind have observed some 
distinct properties in the things which have caused 
them to be singled out, and have that name appro- 
priated to them. There is snch a class of words 
assigned by popular usage to those events (amongst 
others) of which Plrobability takes account. If we 
examine them we shall find, I think, that they direct 
us unmistakeably to the two-fold aspect of the ques-^ 
tion, — ^the objective and the subjective, the quality in 
the events and the state of our minds in consider- 
ing them, — ^that have occupied our attention during 
the former chapters. 

The word < extraordinary,' for instance, seems to 
point to the observed fact^ that events are arranged 
in a sort of ordo or rank. No one of them might 
be so exactly placed that we could have inferred its 
position, but when we take a great many into account 
together, running our eye, as it were, along the line, 
we begin to see that they really do for the most part 
stand in order. Those which stand away £rom the 
line have this divergence observed, and are called 
.extraordinary, the rest ordinary, or in the line. So 
too 'irregular' and 'abnormal' are doubtless used 
from the appearance of things, when examined in 
large numbers, being that of an arrangement by rule 
or measure. This only holds when there fixe a good 



SECT. 31.] THE LOGIC OF CHANCE. 103 

many; we could not speak of single events being 
so arranged. Again, the word 'law/ in its pMlo* 
sophical sense, has now become quite popularized. 
How the term became introduced is not certain, but 
I think there can be little doubt that it was some- 
what in this way : — The observed effect of a law is 
to produce regul^.rity where it did not previously 
exist ; when then a regularity began to be perceived 
in nature, the same word was used, whether the cause 
was supposed to be the same or not. In each case 
there was the same generality of agreement, subject 
to occasional deflection*. 

On the other hand, observe the words ' wonderful,' 
* unexpected,* * incredible.' Their connotation de- 
scribes states of mind simply ; they are not conflned 
to Probability, but mean that the events they denote 
are such as from some cause we did not expect would 
happen, and at which therefore, when they do happen, 
we are surprised. 

Now when we bear in mind that these two classes 
of words are in their origin perfectly distinct ; — ^the 
one denoting simply events of a certain character ; the 
other, though also denoting events, comioting simply 
states of mind; — and yet that they are universally 
applied to the same events, so as to be used as per- 

• This would still hold of empirical laws which may be capa- 
ble of being broken : we now have very much shifted the word, 
to denote an ultimate law which it is supposed cannot be broken. 



104 THE LOGIC OP CHANCE. [CHAP. III. 

fectly synonymous, we have in this a striking illus- 
tration of the two sides under which Probability may 
be viewed, and of the universal recognition of a close 
connection between them. The words are popularly 
used as synonymous, and we must not press their 
meaning too far ; but if it were to be observed, as 
I think it could, that the application of the words 
which denote mental states is wider than that of 
the others, we should have an illustration of what 
has been already observed, viz. that the Province of 
Probability is not so extensive as that over which 
variation of belief might be observed. Probability 
only considers the cases in which this variation is 
brought about in a certain definite statistical way. 

§ 32. It will be found in the end both interest-, 
ing and important to have devoted some attention to 
this subjective side of the question. In the first 
place, as a mere speculative inquiry the quantity of 
our belief of any proposition deserves notice. To 
study it at all deeply would be to trespass into the 
province of Psychology, but it is so intimately con- 
nected with our own subject that we cannot avoid 
all reference to it. We therefore discuss the laws 
under which our expectation and surprise at isolated 
events increases or diminishes, so as to account for 
these states of mind in any individual instance, and, 
if necessary, to correct them when they vary from 
their proper amount 



1 



SECT. 32.] THE LOGIC OP CHANCE. 105 

But there is another more important reason than 
this. It is quite true that when the subjects of our 
discussion in any particular instance lie entirely with- 
in the province of Probability, they may be treated 
without any reference to our belief. We may or we 
may not employ this side of the question according to 
our pleasure. If, for example, I am asked whether it 
is more likely that A. B. will die this week, or that it 
will rain to-morrow, I may calculate the chance 
(which really is at bottom the same thing as my 
belief) of each, find them respectively, one-sixteenth 
and one-seventeenth, say, and therefore decide that my 
'expectation' of the former is the greater, viz. that 
it is the more likely event. In this case the process 
is precisely the same whether we suppose our belief to 
be introduced or not; our mental state is, in fact, 
quite immaterial to the question. But, in other 
cases, it may be different. Suppose that we are com- 
paring two things, of which one is wholly alien to 
Probability, the only ground they have in common 
may be the amount of belief to which they are re- 
spectively entitled. We cannot compare the fre- 
quency of their occurrence, for one may occur too 
seldom to judge by, perhaps it may be unique. It 
ha£i been already said, that our belief of many events 
rests upon a very complicated and extensive basis. 
My belief may be the product of many conflicting 
arguments, and many analogies more or less remote; 



"1 



THE LOGIC OF CHANGS. [cHAP. IIL 

/ proofs themselves may have mostly faded from 
/mind, but they will leave their effect behind them 
i a weak or strong conviction. At the time, there- 
fore, I may still be able to say, with some degree 
of accuracy, though a very slight degree, what amount 
of belief I entertain upon the subject. Now we 
cannot compare things that are heten^eneous ; if, 
therefore, we are to decide between this and a thing 
determined by Probability, it is impossible to appeal 
to chances or frequency of occurrence. The measure 
of belief is the only common ground, and we must 
therefore compare this quantity in each case. The test 
afforded will be an exceedingly rough one, for the rea- 
sons mentioned above, but it will be better than none ; 
in some cases it will be found to tonish all we want. 
Suppose, for example, that one letter in a million 
is lost in the Post Office, and that in any given 
instance I wish to know, which is more likely, that a 
letter has been so lost, or that my servant has stolen 
it ) If the latter alternative could, like the former, 
be stated in a numerical form, the comparison would 
be simple. But it cannot be reduced to this form, at 
least not consciously and directly. Still, if we could 
feel that our belief in the man's dishonesty was 
greater than one-millionth, we should then have 
homogeneous things before us, and therefore compa- 
rison would be possible. 

§ 33. We are now in a position to give a toler- 



SECT. 33.] THE LOGIC OF CHAWCE. 107 

ably accurate definition of a phrase which we have 
frequently been obliged to employ, or incidentally 
to suggest, and of which the reader may have looked 
for a definition already, viz. the probability of an 
event, or what is equivalent to this, the chance of 
any given event happening. I consider that these 
terms presuppose a series ; within the indefinite class 
which composes this series a smaller class is dis- 
tinguished by the presence or absence of some attri- 
bute or attributes, as was fully illustrated and ex- 
plained in a previous chapter. These larger and 
smaller classes respectively are commonly spoken of 
as instances of the 'events' and of 'its happening 
in a given particular way.' Adopting this phrase- 
ology, which with proper explanations is suitable 
enough, I should define the probability or chance 
(I regard the terms as synonymous) of the event 
happening in that particular way as the numerical 
j&action which represents the proportion between the 
two different classes in the long run. Thus, for 
example, let the probability be that of a given in< 
fant living to eighty. The larger series will com- 
prise all men, the smaller all who live to eighty. 
Let the proportion of the former to the latter be 
100 to 1; in other words, suppose that one child 
in a hundred lives to eighty. Then the chance or 
probability that any given child will live to eighty 
is the numerical fraction t^. This assumes that 



108 THE LOGIC OF CHANCE. [CHAP. Ill; 

the series are of indefinite extent, and of the kind 
which we have described as possessing a fixed type. 
If this be not the case, but the series be supposed 
terminable, or irregularly fluctuating, then in so far 
as this is the case the series ceases to be a subject of 
science. What we have to do under these circum- 
stances, is to substitute a series of the right kind 
for the inappropriate one presented by nature, 
choosing it, of course, with as little deflection as pos- 
sible from the observed facts. This is nothing more 
than has to be done, and invariably is done, when- 
ever natural objects are made subjects of strict 
science. 

A word or two of explanation may be added 
about the expression employed above, 'the propor- 
tion in the long run.' The run must be supposed 
to be very long indeed, in fact never to stop. As 
we keep on taking more terms of the series we shall 
find the proportion still fluctuating a little, but its 
fluctuations will grow less. The proportion, in &ct, 
will gradually approach towards some fixed numerical 
value, .. what mathematicians term its limit. This 
fractional value is the one spoken of above. In the 
few cases in which deductive reasoning is possible, 
this fraction may be obtained without direct appeal 
to statistics, from reasoning about the conditions 
under which the events occur, as was explained in 
the second chapter. 



SECT. 33.] THE LOGIC OP CHANCE. 109 

Here becomes apparent the full importance of the 
distinction so frequently insisted on, between the 
actual irregular series before us and the substituted 
one of calculation, and the meaning of the assertion 
(Oh. I. § 14), that it was in the case of the latter only 
that strict scientific inferences could be made. For 
how can we have a 4imit' in the case of those series 
which ultimately exhibit irregular fluctuations] 
When we say, for instance, that it is an even chance 
that an unvaccinated person recovers from the small- 
pox, the meaning of this assertion is that in the long 
run each alternate person attacked by that disease 
does recover. But if we examined a sufficiently 
extensive range of statistics, we might find that the 
manners and customs of society had produced such a 
change in the type of the disease or its treatment, 
that we were no nearer approaching towards a fixed 
limit than we were at first. The conception of an 
ultimate limit in the ratio between the numbers of 
the two. classes in the series necessarily involves an 
absolute fixity of the type. When therefore nature 
does not present us with this absolute fixity, as she 
scarcely ever does except in games of chance (and not 
demonstrably there), our only resource is to invent 
such a series, in other words, as has so often been said, 
to substitute a series of the right kind. 

The above, which I consider to be tolerably com- 
plete as a definition, might equally well have been 



110 THE LOGIC OP CHANCE. [CHAP. III. 

givea in the last chapter. I have deferred it how- 
ever to the present place, in order to connect with it 
at once a proposition involving the conceptions in- 
troduced in this chapter; viz. the state of our own 
minds, in reference to the amount of belief we enter- 
tain in contemplating any one of the events whose 
probability has just been described. Beasons were 
given against the opinion that our belief admitted 
of any exact apportionment like the numerical one just 
mentioned. Still, it was shown that a reasonable 
explanation could be given of such an expression 
as, my belief is ^th of certainty, though it was an 
explanation which pointed unmistakeably to a series 
of events, and ceased to be intelligible unless viewed 
in such a relation to a series. In so &r, then, as 
this explanation is adopted, we may say that our 
belief is in proportion to the above fraction. This 
referred to the purely intellectual part of belief which 
I cannot conceive to be separable, even in thought, 
from the things upon which it is exercised. With 
this intellectual part there are commonly associated 
many emotions. These we can to a certain extent 
separate, and, when separated, can measure with that 
degree of accuracy which is possible in the case of 
other emotions. They are moreover intelligible in 
reference to the individual events. They will be 
found, I think, to increase and diminish in accord- 
ance^ to some extent, with the fraction which repre* 



SECT. 33.] THE LOGIC OP CHANCE. Ill 

sents the scarcity of the event. The emotion of 
surprise does so with some degree of accuracy. 

The above investigation describes, though in a 
very brief form, the amount of truth which appears 
to me to be contained in the assertion frequently 
made, that the fraction of probability represents also 
the fractional part of full certainty to which our 
belief of the individual event amounts. Any further 
analysis of the matter would seem to belong to Psy^ 
chology rather than to Firobability, 



CHAPTER lY. 
THE RULES OF INFERENCE IN PROBABILITY. 

§ 1. In the previous chapter, an investigation was 
made into what may be called, from the analogy of 
Logic, Immediate Inferences. Given that nine men 
out of ten live to forty, what could be inferred about 
the prospect of life of any particular man ? It was 
shown that, although this step was very fer from 
being so simple as it is commonly supposed to be, and 
as the corresponding step really is in Logic, there was 
nevertheless an intelligible sense in which we might 
speak of the amount of our belief in any one of these 
proportional propositions, and justify that amount. 
We must now proceed to the consideration of infer- 
ences more properly so called, I mean inferences of 
the kind which form the staple of ordinary logical 
treatises. In other words, having ascertained in what 
manner particular propositions could be inferred from 
the general propositions which included them, wo 
must now examine in what cases one general proposi- 
tion can be inferred from another. By a general pro- 
position here is meant, of course, a general proposition 
of the statistical kind contemplated in Probability. 



SECT. 3,] LOGIC OF CHAKCE. 113 

The rules of such inference being very few and simple, 
their consideration will not detain us long. 

§ 2. From the data now in our possession we are 
able to deduce the rules of probability given in ordi- 
nary treatises upon the science. It would be, more 
correct to say that we are able to deduce some of these 
rules, for, as will appear on examination, they are of 
two very different kinds, resting on entirely distinct 
grounda They might be divided into those which 
are formal, and those which are merely experimental. 
This may be otherwise expressed by saying that, from 
the kind of series described in the first two chapters, 
some rules will follow necessarily by the mere appli- 
cation of arithmetic; whilst others either depend 
upon peculiar hypotheses, or demand for their esta- 
blishment continually renewed appeals to experience, 
and extension by the aid of Induction. We shall 
confine our attention at present principally to the 
former class ; the latter can only be fully understood 
when we have considered the connection of our sci- 
ence with Induction. 

§ 3. (1) We can make inferences by simple 
addition. If, for instance, there are two distinct pro- 
perties observable in various members of the series, 
which properties do not occur in the same individual ; 
it is plain that in any large batch, the number that 
are of one kind or the other will be equal to the sum 
of those of the two kinds separately. One man in ten, 

8 



114 LOGIC OP CHANCE. [cHAP. IV. 

say, is over six feet in height, and one in twelve is 
nuder £ve. Take a large number, say 120,000, then 
there will be about 12,000 tall and 10,000 short men 
amongst them; obviously therefore those who are of 
one kind or the other will be 22,000 in number. This 
rule, in its general algebraical form, would commonly 
be expressed in the language of Probability as foU 
lows : — If the chances of two incompatible events be 

respectively — and -• the chance of one or other of 

them happening is — + - or . Similarly if 

^^ ^ m n mn 

there were more than two such events. On the prin- 
ciples adopted in this Essay the rule, when thus ex- 
pressed, means precisely the same thing as when it is 
expressed in the statistical form. It was shown, at 
the conclusion of the last chapter, that to say, for ex- 
ample, that the chance of a given event happening in 

a certain way is ^, is only another way of saying that 

it does happen in that way once in six times. 

It is plain that a corollary to this rule might be 
obtained, in precisely the same way, by subtraction 
instead of addition. Stated generally it would be as 
follows : — If the chance of one or other of two incom- 
patible events be — , and the chance of one alone be-, 

the chance of the remaining one will be or . 

° m n mn 



SECT. 4.] LOGIC OF GHAXCIL 115 

Ex. If the chanee of being either ihot or bajon* 
eted in a battle is i, and that of being shot is ^, 
then that of being bayoneted is ^. (Supposkig that 
a man oannot be both shot and bayoneted). 

§ 4. (2) We can also make inferences by mul- 
tiplication. Suppose that the two events, instead of 
being incompatible as in the preyions examples, are 
invariably connected together. Let a certain propor- 
tion of the members of the series possess a given pro- 
perty, and a certain proportion again of these, and 
of these only, possess another property, then the pro- 
portion which possess both properties is found by mul- 
tiplying together the two fractions which represent 
the above two proportions. One man in ten, say, 
is over six feet in height^ and one in fifty of these 
tall men, and of them only, has red hairj then, of 
the men whom we casually meet, about one in 500 
will be tall and red-haired. 

This rule is variously expressed in the language of 
Probability; perhaps the following is the commonest 

form: — If the chance of one event is — , and the 

m 

chance that if it happens another will also happen is 

-, the consequent chance of the latter is — . 

The above inferences are necessary, in the sense in 
which arithmetical laws are supposed to be necessary, 
and they do not demand for their establishment any 

8—2 



116 LOGIC OF CHANCE. [CHAP. IV. 

arbitrary hypotJiesis. "Wo assume in them no more 
than is warranted by the data actually given to us, and 
make our inferences from these data by the help of 
arithmetic. The formula, however, which we are 
about to examine next stands on a somewhat different 
footing. 

§ 5. (3) In the two former rules we considered 
cases in which the data were supposed to be given 
under the conditions that the properties which distin- 
guished the different kinds of events whose frequency 
we discussed, were respectively known to be discon- 
nected and known to be connected. Let us now sup- 
pose that no such conditions are given to us. One 
man in ten, say, has red hair, and one in twelve stam- 
mers; what conclusions could we then draw as to the 
chance of any given man having one only of these two 
attributes, or neither, or both? It is clearly possible 
that the properties in question might be inconsistent 
with one another, so as never to be found combined in 
the same'person; or all the stammerers mi^t have 
red hair; or the properties might be allotted* in 
almost any proportion whatever. If we are perfectly 

* I say, almost any proportion, because, as may easily be seen, 
arithmetic imposes certain restrictions upon the assumptions that 
can be made. We could not, for instance, suppose that all the 
red-haired men are stammerers, for in any given batch of men the 
former are more numerous. But the range of these restrictions 
is limited, and their existence is not of importance in the above 
discussion. 



SECT. 5.] LOGIC OP CHANCE. 117 

ignorant upon these points, it would seem that no 
inferences whatever could be drawn about the re- 
quired chances. 

Inferences however a/re drawn. An escape from 
,the apparent indeterminateness of the problem, as 
above descnbed, is found by assuming that, not merely 
will one-tenth of the whole number of men have red 
hair (for this was given as one of the data), but also 
that one-tenth alike of those who do and who do not 
stammer have red hair. Let us take a batch of 1200, 
as a sample of the whole. Now, from the data which 
were originally given to us, it will easily be seen that 
in every such batch there will be on the average 120 
who have red hair, and therefore 1080 who have not. 
And here by rights we ought to stop, at least until we 
have appealed again to experience; but we do not 
stop here. From data which we have manufactured 
for ourselves we go on to infer that of the 120, 10 (t. e, 
one-twelfth of 120) will stammer, and 110 (the remain- 
der) will not. Similarly we infer that of the 1080, 90 
stammer, and 990 do not. On the whole, then, the 
1200 are thus divided: — ^red-haired stammerers, 10; 
stammerers without red hair, 90; red-haired men who 
do not stammer, 110; men who neither stammer nor 
have red hair, 990. 

This rule, expressed in its most general form in 
the language of Probability, would be as follows :— 
If the chances of a thing being p and q are respect- 



118 LOGIC OF CHANCB. [CHAP. IV. 

ively — - and - , then the chance of its being both p 
'' m n . 

and a is — , p and not q is , q and not p is 

^ mn ^ mn ^ ^ 

w - 1 . , . . fe» - 1) (» - 1) - 
, not p and not q is ^ ^-^ -', where p 

and q are independent. 

§ 6. The assumption in the last section is there 
given in its most glaring form. I cannot but think 
however that most writers on the subject do implioiliy 
adopt it as it there stands, implying that where we 
know nothing about the distribution of the propertaes 
alluded to we must assume them to be distributed as 
above described, and therefore apportion our belief 
in the same ratio. This is called 'assuming the events 
to be independent^' the supposition being made that 
the rule will certainly follow from this independence, 
«nd that we have a right, if we know nothing to the 
contrary, to assume that the events are independent 

The validity of this last claim has already been 
discussed in the fbst chapter; it is only another of 
the attempts to construct ^ 'priori the series which 
experience will present to us, and one for which no 
such strong defence can be made as for the equalily 
of heads and tails in the throws of a penny. But the 
meaning to be assigned to the 'independence' of the 
events in question demands a moment's consideration. 

The circumstances of the problem are these. Ther^ 



8SCT. 7.] LOGIC OF CHAKC& 119 

are two different qualities, by the presence or absence 
of which amongst the individuals of a series two 
distinct pairs of classes of these individuals are pro- 
duced. For the establishment of the rule under dis- 
cussion it was found that one supposition was both 
necessary and sufficient, namely, that the division 
caused by each of the above distinctions should sub- 
divide each of the classes in the other pair in the 
same ratio in which it subdivides the whole. If the 
independence be granted and so defined as to mean 
this, the rule of course will stand, but, without espe- 
cial attention being drawn to the point, it does not 
seem that the word would naturally be so understood. 
§ 7. The above are the prlnc^al fundamental 
rules of inference which the science can give us. A 
few remarks may now be added about the form which 
they assume in some other wcnrks upon the subject. 
Reference has already been made to Profbssor Pe 
Morgan's assertion* about the province of Probabi- 
lity, that it haa to study '^the effect which partial 
belief of the premises produces with respect to the 
conclusion," whereas in ordinary logic we suppose the 
premises to be absolutely true. This will be the fittest 
place for explaining clearly my resui(»is for differing 
from him. Let us xecur to the first of the examples 
quoted in this chapter; it was as follows: — One man 
in ten is over six feet high, one in twelve is under 
* I>0 Morgan's SomuU Logic, Pref aoe^ p. v. 



126 LOGIC OF CHANGE. [CHAP. IT. 

fire; from this we inferred that eleven in sixty were 
not between five and six feet. These propositions, 
when stated in the form of a chance^ would be ex* 
pressed as follows. The chance of a man being over 
six feet is ^, that of his being under ^ve is ^ ; 
therefore the chance of his not being between thesQ 
heights is H. It has been stated, and fully ex- 
plained, that these two forms of assertion mean pre- 
cisely the same thing. 

But it was also shown that there was a subjective 
side of the question, in accordance with which these 
propositions might assume the following form. My 
belief that a man will be over six het is represented 
hy tV» There is no need to recur to this beyond 
reminding the reader, that a proposition of this kind 
only became intelligible or capable of justification 
when viewed in connection with the statistical &cts 
to which it referred. Now Professor De Morgan 
seems to hold that these propositions, in the latter 
form, viz. in the form of statements of partial belief, 
can be inferred one from the ofcher. To me it seems, 
on the contraiy, that but little meaning and certainly 
no security can be attained by so regarding the pro- 
cess of inference. These probabilities must first be 
supposed to be re-translated into statem^its about 
the things, and then the inferences must be drawn 
from observations upon these things. This part of 
the operation is carried on, as already shown, by the 



9ECT. 8.] LOGIC OF CHANCE. 121 

ordinary rules of arithmetic. The oonolusion, when 
obtained, may, of course, be stated in the subjective 
form, equally with the premises; but it is difficult to 
see how the process of inference can be conceived as 
taking place in that form. Certainly no proof of it 
can then be given. If therefore the process of in- 
ference be so expressed it must be regarded as a sym- 
bolical process, symbolical of such an inference about 
things as has been described above, and it thei*efore 
seems to me more advisable to examine it under this 
latter form. 

§ 8. The above, then, being the fundamental 
rules of inference in Probability, the question at once 
arises, What is their relation to the great body of 
formulsB which are made use of in treatises upon the 
science, and in practical applications of it) Tho 
reply would be that these formulse, in so far as they 
properly belong to the science, are nothing else in 
reality than applications of the above fundamental 
rules. Such applications may assume any degree of 
complexity, for owing to the difficulty of particular 
examples, in the form in which they actually present 
themselves, recourse must sometimes be made to the 
profoundest theorems of mathematics. Still we ought 
not to regard these theorems as being anything else 
than oonvenient and necessary abbreviations of arith- 
metical processes, which in practice have become too 
xumbersome to be otherwise performed. 



122 LOGIC OF GHAKCE. [CHAP. IV. 

§ 9. This explanation, will account for some of 
the rules as they are ordinarily given, but by no 
means for all of them. It will account for those 
which are demonstrable by the certain laws of arith- 
metic, but not for those which in reality rest only 
upon inductive generalizations. And it can hardly 
be doubted that many rules of the latter description 
have become associated with those of the former, so 
that in popular estimation they have been blended 
into one system, of which all the separate rules are 
supposed to possess a similar origin and equal cer- 
tainty. Hints have already been frequently given of 
this tendency, but the subject is one of such extreme 
importance that a separate chapter must be devoted 
to its consideration. 

§ 10. Before concluding this chapter the reader 
is reminded, in order to prevent misapprehension, 
that no ajssumption is made in the above remarks 
about the nature of demonstrative truth as involved 
in the rules of arithmetic. We have called them ne- 
cessary rules, but it is quite immaterial for our pre- 
sent purpose whether they be derived from experience 
or not. The most strenuous assertor of their experi- 
mental origin will not deny that, as things now are, 
they ai*e sharply marked offf from mere inductive 
generalizations in respect of the strength of our con- 
victions about their invariable truth. With oar 
present mental constitution and experience the former 



SECT. 10.] LOGIC OF CHAKCB. 123 

are irreyersible and the latter generallj are not, and 
this is abundantly sufficient to classify them apart. 
The discussion of such a question as this belongs, as 
do many other discussions, to the science of the laws 
of evidence and discovery in their most general form, 
rather than to such a limited portion of them as vre 
are now occupied with. 



CHAPTER V. 

GENERAL REMARKS ON THE RESULTS OF THE 
FOREGOING CHAPTERS. 

§ 1. As the remarks in the present chapter will be 
of a somewhat general character, it will be advisable 
to pause for a moment in order to obtain a dear con- 
ception of our present standing-point. 

On the objective side, then, we have a series of 
events occurring in any order in time. This series of 
events is at bottom nothing but a series of groups of 
substances and attributes, to which groups various 
other attributes are found united, in a certain definite 
proportion of cases out of the whole. The existence 
of such a series is supposed to be known; by what 
evidence it may be established in any particular in- 
stance we are not called upon at present to enquire. 

With regard to the subjective side, we must sup- 
pose some person, say myself, contemplating this series. 
I mentally single out some one or more of the indi- 
viduals which compose the series, and endeavour to 
form an opinion, judging solely by the statistical fre- 
quency with which the attributes occur, whether in 
these selected instances the occasional attributes will 



SEC?r. 2.] LOGIC OF CHANCE. 125 

be present or not. There does not seem to be any 
better mode of expressing this than to say, that I form 
a coTiceptwn or anticipation of some member of the 
series at present unknown to me, unknown at least in 
some of its characteristics ; my conception includes in 
it, or excludes from it as the case may be, the occa- 
sional attributes, and in this respect I cannot of course 
feel certain about its being a correct representation* of 
the facts. It is the duty of Probability to investigate 
with what degree of strength this conception should 
be entertained,, in other words, how firmly we ought 
to believe it to be correct. 

§ 2. Sometimes the member of the series thus 
singled out for anticipation may be supposed to be 
already within the certain grasp of experience in re- 
gard to some of its characteristics, our doubt and there- 
fore the possible inaccuracy of our conception referring 
only to the remaining characteristics; sometimes it 
may be altogether unknown as yet, except as occupy- 
ing a certain numerical position in the succession. In 
every case, however, we shall find that there is present 
to our minds a conception of an event which is at 
present tinged with doubt and which we are waiting 
to confirm or reject; Probability refers to this time 
of pause and doubt. For example, I know that four 
children in ten live to be fifty. Here the series is 
one of children, to four out often of whom we are able 
to assign the propei-ty of living to fifty. T select one, 



126 LOGIC OS* GHAKOB. [CHAP. Y. 

my only remaining doubt is whether it will Hto to 
fifty. This may be expressed by saying that I form 
the conception of it as living to fLffcy, and want to 
ascertain how firmly I should entertain the concep- 
tion. Or the child may as yet be no subject of expe- 
rience, from its not being at presMit in existence. The 
child may be determined simply from its being, say, 
the first bom next week. The process is precisely 
the same in all these instances. A conception is 
formed, and a value (ue. amount of belief as explained 
in Oh. in.) assigned to it solely on statistical grounds. 
§ 3. The above mentioned view of the subject is, 
I apprehend, the ordinary one involved in what is 
sometimes termed Material Logic. This view is not 
indeed so prominently brought before us there, bat 
a little consideration will show that substantially the 
same view is involved in every science which professes 
to draw inferences about external things. In every 
such science we must suppose a certain number of 
facts given in experience, and therefore a certain 
number of propositions known to be true; the object 
aimed at in the inferences is to add to this domain of 
&ct in every direction. It is not easy to see how 
this can be done except by forming conceptions, and 
then ascertaining whether these correspond to &ct 
or not. It is true that this process is obscured in the 
case of those ordinary inferences which are supposed 
to amount to demonstration, for here the same infer- 



SBCT. 3.] LOGIC OF CHAKCE. 127 

ence whicli first suggests the conception to ns may be 
the very thing which assures us of its truth. If so, 
the conception may be described as springing at once 
out of non-existence into the domain of fact. But 
whenever we are drawing conclusions about things by 
means of inductive rules which do not amount to de- 
monstration, especially when the fact to be established 
depends upon a combination of several such argu- 
ments, we shall hardly be able to avoid taking the 
view now under discussion. In all such cases we 
have a multitude of conceptions (or whatever other 
name we may give to these notions in our minds) 
which we should be unwilling to call imaginary, and 
yet which we should scarcely be able, on the other 
hand, to speak of as &cts. They are rather in a sort 
of noviciate, and qualifying for facts. But they are 
certainly at that moment present to us, and so far 
really existent in the mind. Our position, therefore, 
in these cases seems distinctly that of entertaining a 
conception, and the process of inference is that of 
ascertaining to what extent we are justified in adding 
this conception to the already-received body of truth 
and fact. This view of the subject is for more forcibly 
set before us in Probability than in any of the Induc- 
tive sciences, owing to the fact that in Pi-obability we 
distinctly take notice of, and regard as evidence, rea- 
sons so faint that they would scarcely be called by 
any other name than mere hypotheses elsewhere. But 



128 LOGIC OF CHANCE. [CHAP. V. 

however slight may be the statistical grounds for be- 
lieving in a thing, these grounds certainly suggest the 
conception of it to the mind, and they give some force 
which the mind can appreciate for believing in its 
truth. 

§ 4. For additional clearness two brief remarks 
may be added. Let it be observed then that this is 
in no sense an adoption of the Conceptualist view of 
Logic. It would be so were we to set before us as 
our object to ascertain whether, for example, the con- 
ception * dying after fifty years' is or is not involved 
in that of * being bom,' or with what amount of force 
we should believe it to be so contained, (were this 
last expression quite intelligible). But it is a very 
different thing to make out whether the former con- 
ception is or is not trtie, that is, whether it does or 
does not fit in with the rest of our experience about 
men. This latter is inference about things, and it is 
in this sense that Probability is underatood in this 



But, at the same time, when we speak about con- 
verting our conceptions into matters of fact, we do 
not at all imply any opinion as to whether these mat- 
ters of fact are not at bottom resolvable into a col- 
lection of subjective impressions. This is a question 
with which we, as a kind of logicians, are in no way 
concerned. I may con-ect a person's impression of a 
steam-engine, for instance, or tell him that it is a 



SECT. 5.] LOGIC OP CHANCE. 129 

false one, without committing myself to any assertion 
as to whether all our experience of steam-engines can 
bring us to anything more at bottom than subjective 
impressions. 

Keeping the foregoing remarks in mind, we shall 
easily see our way to several useful inferences. 

§ 5. In the first place it will be seen that in 
Probability time has nothing to do with the ques- 
tion; in other words, it does not matter whether the 
event, whose probability we are discussing, be past, 
present, or future. The question, in its simplest form, 
is this: — Statistics (extended by Induction) inform 
us that a certain event has happened, does happen, 
or will happen, a certain proportion of times in a 
certain way. We form a conception of that event, 
and regard it ag^ossible; but we want to do more; 
we want to know (from statistical data alone of 
course) how mwcA we ought to expect iti (under the 
explanations ah*eady given about quantity of belief). 
There is therefore a sort of relative futurity about the 
event, inasmuch as our knowledge of the fact, and 
therefore our justification or otherwise of the correct- 
ness of our surmise, almost necessarily comes after 
the surmise was formed; but the futurity is only 
relative. The evidence by which the question is to 
be settled may not be forthcoming yet, or we may 
have it by us but only consult it afterwards. It is 
fipom the feet of the futurity being, as above described, 

9 



130 LOGIC OP CHANCE. [CHAP. V. 

only relative, that I liave preferred to speak of the 
conception of the event rather than of the anticipa- 
tion of it. The latter term, which in some respects 
would have seemed more intelligible and appropriate, 
is open to the objection that it does rather, in popular 
estimation, convey the notion of an absolute as op- 
posed to a relative futurity. 

For example ; a die is thrown. Once in six times 
it gives ace; if therefore we assume, without exami- 
nation, that the throw is ace, we shall be right once 
in six times. In so doing we may, according to the 
usual plan, go forwards in time ; that is, form our 
opinion about the throw beforehand, when no on© 
can tell what it will be. Or we might go hcbchwarda; 
that is, form an opinion about dice that had been, 
cast in time past, and then correct our opinion by 
the testimony of some one who had been a witness 
of the throws. In either case the mental operation 
is precisely the same ; an opinion formed merely on 
statistical grounds is afterwards corrected by specific 
evidence. The opinion may have been formed upon 
a past, present, or future event; the evidence after- 
wards may be our own eye-sight, or the testimony of 
others, or any kind of inference; by the evidence is 
merely meant the subsequent examination of the case 
that is assumed to set the matter at rest It is quite 
possible, of course, that this specific evidence should 
never be forthcoming; the conception in that case 



w^ 



SECT. 6.] LOGHC OP CHANCE. 131 

remains as a conception, and never obtains tHat degree 
of conviction which qualifies it to be regarded as a 
'&ct.' This is the case with all past throws of dice, 
the result of which have not been recorded. 

In discussing games of chance there are obvious 
advantages in confining ourselves to what is really, as 
well as relatively, future, for in that case direct infor- 
mation concerning the contemplated result being im- 
possible, all persons are on precisely the same footing 
of ignorance, and must form their opinion entirely 
from the frequency of occurrence of the event in 
question. On the other hand, if the event be past, 
there is almost always evidence of some kind and of 
some value, however faint, to inform us what the 
event really was; if this evidence is not actually at 
hand, we can generally, by waiting a little, obtain 
something that shall be at least of use to us in form- 
ing our opinion. Practically therefore we generally 
confine ourselves, in anticipations of this kind, to 
what is really future, and so in popular estimation 
futurity becomes indissolubly associated with proba- 
bility. 

§ 6. But there is an error closely connected with 
this^ or at least an inaccuracy of expression constantly . 
leading to error, which has found large acceptance, 
and has been sanctioned by some writers of the great- 
est authority. Both Bishop Butler and Mr Mill 
have drawn attention to the distinction between im- 

9—2 



132 LOGIC I OP CHANCE. [CHAP. 

probability before the event and improbability after 
the event, which they assert to be perfectly different 
things; if however the principles laid down above 
be correct, such a distinction as this cannot be main- 
tained. 

Butler's remarks on this subject occur in his 
Analogy^ in the chapter on miracles. Admitting 
that miracles are very improbable he strives to obtain 
assent for them by showing that other events, which 
are also very improbable, are received upon what is 
in reality very slight evidence. He says, '* There is 
a very strong presumption against common speculative 
truths, and against the most ordinary facts, before the 
proof of them; which yet is overcome by almost any 
proof. There is a presumption of millions to one 
against the story of Caesar, or of any other man. 
Eor suppose a number of common facts so and so 
circumstanced, of which one had no kind of proo^ 
should happen to come into one's thoughts, every one 
would without any possible doubt conclude them to 
be false. And the like may be said of a single com- 
mon fact." 

It surely needs but little reflection to see that his 
illustration of his position completely overturns it. 
For is he not in reality speaking of two perfectly dis- 
tinct things here? In the ' improbable thing before the 
proof we have represented to us a man * thinking of 
the story of Caesar,' that is, forming a conception of 



er I 



SECfT. 7.] LOGIC OF CHANCE. , 133 

certain historical events, mthout amy grounds, and 
speculating as to what value is to be attached to the 
probability of its truth. Such a conception is of 
course, as he says, rejected as utterly improbable. 
Now what does he understand by the * improbability 
after the proof '1 That a story not adopted at ran- 
dom, but actually suggested and supported by wit- 
nesses, should be true. This latter might be accepted ; 
the former would undoubtedly be rejected; but all that 
this proves, or rather illustrates, is that the testimony 
of almost any witness is vastly better than a mere 
guess. We may in both cases alike speak of *the 
event' if we will; but it should be clearly understood 
that what is really present to the man's mind, and 
what is to have its probable value assigned, is the 
conception of an event; and surely no two conceptions 
can have a much more important distinction put be- 
tween them than that which is created by supposing 
one to be an unsupported guess, and the other the 
report of witnesses. 

§ 7* Mr Mill, in a chapter of his Logic on the 
Grounds of Disbelief, speaks of persons making the 
mistake of'* overlooking the distinction between (what 
may be called) improbability before the feet, and im- 
probability after it, two different properties, the 
latter of which is always a ground of disbelief, the 
former not always." He instances the throwing of 
a die. It is improbable beforehand that it should^ 



134 LOGIC OP CHANCE. [CHAP. V. 

tiim up ace, and yet afterwards, " there is no reason 
for disbelieving it if any credible witness asserts it/* 
The introduction of the sentence, *if any credible 
witness asserts it,* alters the whole question. So with 
his other example; Hhe chances are greatly against 
A, B,'a dying, yet if any one tells us that he died 
yesterday we believe it.' 

That the amount of our beliei^ in the above cases, 
has no necessary connection with the feet of the event 
being one which has already happened, or, as it is ex- 
pressed, of the probability being after the fact^ seems 
23lain. Conceive for a moment, that some one had the 
power of knowing whether A, B, would die or not, (he 
might have some secret sources of knowledge unknown 
to ourselves). If he told us that A. B, would die to- 
morrow we should in that case be precisely as ready 
to believe him as when he tells us that A. B, has died. 
If we continued to doubt it would simply be because 
we thought that with him, as with us, the assertion 
rested on a guess and nothing more. So with the event 
when past ; the fact of its being past makes no difference; 
until this credible witness tells us, we should doubt it 
if it came into our minds just as much as if it were 
future. 

There is precisely the same distinction to be 
drawn in these examples as in those of Bishop Bailer. 
What is really pi^sent to the man's mind is in one 
case a groundless conjecture (grounded only, that is, 



■I^P^^^PJP 



flECT. 8.] LOGIC OF CHANCE. 135 

on statistical information about the average), in the 
other the statement of a witness. The observer has 
in each case to assign its due value to the conception, 
and the conceptions being obtained in such different 
ways will naturally be valued differently. 

§ 8. Butler's general argument in the chapter in 
question has been a good deal criticized. But his 
extraordinary opinion that every particular event is, 
when we come to think of it, excessively improbable, 
lias attracted comparatively little notice. Connecting 
it with his other assertion, that these events are never- 
theless established by slight evidence, we are forced to 
one of two alternatives* Either we are every day 
believing things which we have no grounds to believe, 
(this is adopting the common signification of the word 
improbable, as being nearly equivalent to deserving of 
little belief). Or on. the other hand we must admit 
that the improbability of an event has little or no 
connection with the degree of our belief of it. The 
former alternative would almost effect a revolution in 
our belief and the latter in our language. 

It was apparently to avoid such a dilemma as this 
that Mr Mill has insisted upon the distinction between 
the probability before and after happening. He admits 
that the event would be improbable beforehand, but 
denies that it is so afterwards. Butler on the other 
hand admitted the event in both cases to be improbable, 
and yet claimed that in one case it could be easily 



136 LOGIC OP CHANCE. [cHAP. ▼. 

proved, with the object of course of obtaining equally 
easy credence for it in the other. 

§ 9. If we bear in mind the distinction explained 
at the commencement of this chapter, we may see our 
way to a simple and satisfactory solution of the diffi- 
culty. We must remember that in strictness it is 
not an event which is improbable ; it is to our con- 
ception of it only, or to the story in which the con- 
ception is conveyed, that this epithet can be applied ; 
an event in itself can only be imcommon. I will 
refer for illustration to another example quoted by 
Mr Mill from Laplace. There is a lottery with 1000 
tickets; it is therefore 999 to 1 against any particular 
number, say 79, being drawn. But now a witness 
whose veracity is but small, say t^, comes and tells us 
that 79 has been drawn. By saying that a man's 
veracity is ^ is meant that one in ten only of his 
statements are true. We conclude, therefore, on 
principles discussed in the preceding chapter, that the 
chance that he is speaking truth in this case is ^. Here 
therefore it seems that we have found an instance in 
which an exceedingly improbable event is rendered 
moderately probable by means of the testimony of a 
witness of no extraordinary veracity. For it will 
most likely be maintained that the event in each case 
is precisely the same, namely the drawing of No. 79. 

§ 10. But let us look a little closer. We shall 
then see that what is really improbable' in the former 



1 



SECT. 11.] LOGIC OF CHANCE. 137 

case is our conception, that is, our guess, about the 
No. 79. We call it improbable in the first case because 
we are convinced that once only in a thousand times 
will such a guess be found to be correct. In the second 
case, also, the improbable thing is our conception about 
79, but it here comes to us not as a guess but con- 
veyed by a witness of given veracity. The ground it 
has to be called improbable is of the same kind as 
before, viz. because once only in ten times will it be 
found to be correct. But the mode in which the 
conception is obtained alters the amount of impro- 
bability exceedingly. Before, it was 999 to 1 against 
its being correct, now it is only 9 to 1. 

§ 11. The distinction between these writers 
seems to be that in Mi* Mill there is little more than 
an inaccuracy of expression; it does not appear that 
any directly erroneous inference has been made. 
Butler, however, meant exactly what he said; he was 
evidently satisfied with his principle, for he appeals 
to it again in his Analogy, He seems really to have 
believed that any proposition which was wildly im- 
probable beforehand, was to be adopted afterwards the 
moment it was testified to by a generally trustworthy 
witness. He does not distinguish between a guess 
and an observation. To me the distinction between 
probability before and after the fact seems to resolve 
itself simply into this; — ^Before the fact we often have 
no better means of information than to guess one of 



138 LOOIO OF CHANCE. [CHAP. T. 

several possible alternatives, after the fact we often 
have, in addition to the guess, specific evidence; hence 
our estimate in the latter case is generally of more 
value. But if these characteristics were inverted, if, 
that is, we were to confine ourselves to guessing about 
the past, and if we could find any additional evidence 
about the future, the respective values of the estimates 
would also be inverted. The difference of these values 
has no connexion with time, but depends entirely 
upon the different grounds upon which our conception 
of the event in question rests. 

§ 12. The origin of the mistake just discussed is 
worth enquiring into. I take it to be as follows. It 
is often the case as above remarked, when we are specu- 
lating about a future event, and almost always when 
that future event is a game of chance, that all persons 
are in precisely the same condition of ignorance in 
respect to it. The limit of available information is 
confined to statistics, and amounts to the knowledge 
that the unknown event must assume some one of 
various alternative forms. The conjecture therefore 
of any one man about it is as valuable as that of any 
other. But in regard to the past the case is very dif- 
ferent. Here we are not in the habit of relying upon 
statistical information. Hence the conjectures of dif- 
ferent men are of extremely different values; in the 
case of many they amount to what we call positive 
knowledge. This puts a broad distinction, in popular 



1 



SECT. 13.] LOGIC OP CHANCS. 139 

estimation, between what maj be called the objective 
certainty of the past and the future, which from the 
standing-point of a science of inference ought to have 
no existence. 

In consequence of this, when we apply to the past 
and the future respectively, the somewhat ambiguous 
expression Hhe chance of the event' it commonly 
comes to bear very different significations. Applied 
to the future it bears its proper meaning, namely, the 
value to be assigned to a conjecture upon statistical 
grounds. It does so, because in this case hardly any 
one has more to judge by than such conjectures. But 
applied to the past it shifts its meaning, owing to the 
fact that whereas some men have conjectures only, 
others have positive knowledge. By the chance of 
the event is now often meant, not the value to be 
assigned to a conjecture founded on statistics, but to 
such a conjecture derived from and enforced by any 
body else's conjecture, that is by his knowledge and 
his testimony. 

§ 13. There is a class of cases in apparent oppo- 
sition to some of the statements in this chapter, but 
which will be found, when examined closely, to con- 
firm them in a remarkable manner. I am walking, 
say, in a remote part of the country and suddenly 
meet with a Mend. At this I am naturally surprised. 
Yet if the view be correct that we cannot properly 
speak about events in themselves being probable or 



140 LOGIC OF CHANCE. [CHAP. V: 

improbable, but only of onr conjectures about them, 
how do we explain this ? We had formed no conjec- 
ture beforehand, for we were not thinking about 
anything of the kind, but yet few would fieiil to feel 
surprise at such an incident. 

The reply might fairly be that we had formed 
such anticipations tacitly. On any such occasion 
every one unconsciously divides things into those 
which are known to him and those which are not. 
During a considerable previous period a countless 
number of persons had met us, and all fallen into the 
list of the unknown to us. There was nothing to 
remind us of having formed the anticipation or dis- 
tinction at all, until it was suddenly called out into 
vivid consciousness by the exceptional event. The 
words we should instinctively use in our surprise seem 
to show this: — *Who would have thought of seeing 
you here V viz. Who would have given any weight to 
the latent thought if it had been called out into con- 
sciousness beforehand 1 We put our words into the 
past tense, showing that we have had the distinction 
lurking in our minds all the time. We always have 
a multitude of such ready-made classes of events in 
our minds, and when a thing happens to Ml into one 
of those classes which are very small we cannot help 
noticing the &tct 

Or suppose I am one of a regiment into which a 
shot flies, and it strikes me, and me only. At this 



SECT. 14.] LOGIC OF CHANCE. 141 

I am surprised, and whyl Our common language 
will guide us to the reason. ^How strange that it 
should just have hit me of all men !' We are thinking 
of the very natural two-fold division of mankind into^ 
ourselves, and every body else ; our surprise is again, 
at it were, retrospective, and in reference to this divi- 
sion. !N'o anticipation was distinctly formed, because 
we did not think beforehand of the event, but the 
event, when it has happened, is at once assigned to 
its appropriate class. 

§ 14. This view is confirmed by the following 
considerations. Tell the story to a friend, and he will 
be a little suiprised, but less so than we were, his 
division in this particular case being, — his friends (of 
whom we are but one), and the rest of mankind. It 
is not a necessary division, but it is the one which 
will be most likely suggested to him. 

Tell it again to a perfect stranger, and his division 
being different (viz. we falling into the majority) we 
shall fail to make him perceive that there is anything 
at all remarkable in the event. 

I am not of course attempting in these remarks 
to justify our surprise in every case in which it exists. 
Different persons might be differently affected in the 
cases supposed, and the examples are therefore given 
mainly for illustration. Still on principles already dis- 
cussed (Ch. lu. § 30) we might expect to find something 
like a general justification of the amount of surprise. 



142 IiOGIC OP CHANCE. [CHAP. 

§ 15. The answer commonly given in these cases, 
is confined to attempting to show that the surprise 
should not arise, rather than showing how it arises. It 
takes the following form, — * You have no right to be 
surprised, for nothing remarkable has really occurred. 
If this particular thing had not happened something 
equally improbable must. If the shot had not hit you 
or your friend, it must have hit some one else who 
was d, priori as unlikely to be hit.' 

For one thing this answer does not explain the 
fact that almost every one is surprised in such cases, 
and surprised somewhat in the different proportions 
mentioned above. So universal a tendency at least 
deserves to be accounted for; I have not seen any 
but that offered above that attempts to account 
for it. 

But again, the answer has the inherent unsatis- 
feictoriness of a dilemma. It admits that something 
improbable has really happened, but gets over the 
difficulty by saying that all the other alternatives were 
equally improbable. A natural inference from this is 
that there is a class of things, in themselves really 
improbable, which can yet be established upon very 
slight evidence. Butler accepted this inference, and 
worked it out to the extraordinary conclusion given 
above. Mr Mill attempts to avoid it by the consider- 
ation of the very different values to be assigned to 
improbability before and after the event. Some fur- 



J 



1 



SECT. 16.] LOGIC OF CHANCE. 143 

ther illustrations of this error will be found in the 
chapter on jBeiUacies. 

§ 16. In connection with the subject at present 
under discussion we will now take notice of a dis- 
tinction which we shall often find insisted on in works 
on Probability, but to which apparently needless im- 
portance has been attached. It is frequently said that 
probability is relative, in the sense that it has a dif- 
ferent value to different persons according to their 
respective information upon the subject in question. 
For example, two persons, A and £, are going to draw 
a ball from a bag containing 4 balls, A knows that the 
balls are black and white, but does not know more; 
B knows that three are black and one white. It will 
be said that the probability of a white ball to ^ is ^, 
and to ^ i. 

But surely there is nothing more in this than the 
principle, equally true in every other science, that our 
inferences will vary according to the data we assume. 
We might just as well speak of the area of a field or 
the height of a mountain being relative, and therefore 
having one value to one person and another to another. 
The real meaning of the example cited above is this; 
A supposes that he is choosing white at random out 
of a series which in the long run gives white and black 
equally often; B supposes that he is choosing white 
out of a series which in the long run gives three black 
to one white. By the application, therefore, of a 



144 LOGIC OF CHANCE. [CHAP. V. 

precisely similar rule they draw different conclusions; 
but so they would under the same circumstances in 
any other science. If two men are measuring the 
height of a mountain, and one supposes his base to be 
1000 feet, whilst the other takes it to be 1001, 
they would of course form different opinions about 
the height. The science of mensuration is not sup- 
posed to have anything to do with the truth of the 
data, but assumes them to have been correctly taken; 
why should not this be equally the case with Pro- 
bability? 

§ 17. The former example, that of the balls and 
bag, appears plausible owing to the fact that two 
different persons, who had not looked into the bag, 
really might form different opinions about its contents. 
But if we take another example, in which the data 
(Eire less mistakeable, we shall see how needless the 
assertion of the relativity of the probability becomes: 
And in most legitimate applications of Probability the 
data offer no more opportunity for difference of opinion 
than do those of any other science. A die, for ex- 
ample, is going to be tossed up. A supposes it to have 
six faces, B only ^ve. Would it not seem somewhat 
frivolous to say that the probabilities of ace to the two 
men are respectively | and ^ ? The reply would be 
that we must at least assume them to have taken pains 
to arrive at correct data, and that a science can- 
not be called upon formally to recognize the erroneous 



SECT. 17.] LOGIC OP CHANCE. 145 

or groundless opinions of the obsei-vers. They must 
take this risk upon themselves. If Probability be con- 
fined to its proper province, no such distinction as the 
above would ever be needed or demanded; for in 
that case all persons may obtain the same statistical 
information if they choose to take the trouble, there- 
fore knowledge inferior to this is not wanted. And 
they can none of them obtain anything more than 
these statistics, therefore superior knowledge is ex- 
cluded. In other words, we shall naturally assume 
the observers to be in this, as in other sciences, all of 
them equally well-informed. If they are not, it is 
their own fault. 

To describe two persons looking at the same bag, 
and to insist upon the different expectations which 
they entertain, and are bound on philosophical grounds 
to support, as to what will come out of it, is to make 
one of those too numerous applications of the theory 
of Probability which have served to bring an un* 
deserved contempt upon the whole science. 



10 



1 



CHAPTER VL 
THE RULE OF SUCCESSION. 

§ 1. In a former chapter we discussed at some 
length the nature of that kind of inference in Proba- 
bility which corresponds to one class of those termed 
in Logic immediate inferences. We ascertained what 
was the meaning of saying, for example, that the 
chance of any given man A, B, dying in a year is J, 
when concluded from the general proposition that one 
man out of three in his circumstances dies. But to 
stop at this point would be to take a very imperfect 
view of the subject. If Probability is a science of real 
inference about things, it must surely give us some- 
thing more than immediate inferences; we must be 
able, by means of it, to step beyond the limits of what 
has been actually observed, and to draw conclusions 
about what is as yet unobserved. This leads at once 
to the question. What is the connection of Probability 
with Induction? This is a question into which it 
will be necessary to enter now with some minuteness. 
§ 2. That there is a close connection between 



SECT. 3.] LOGIC OP CHANCE. 147 

Probability and Induction, must have been observed 
by almost every one who has treated of either subject; 
I have not however seen any account of this connection, 
that seemed to me to be satisfactory. An explicit de- 
scription of it should rather be sought in treatises 
upon the narrower subject, Probability, but it is 
precisely here that the most confusion is to be found. 
The province of Probability being somewhat narrow, 
incursions have been constantly made from it into the 
adjacent territory of Induction. In this way, amongst 
the arithmetical rules discussed in the last chapter 
but one, others have been introduced which, as I shall 
hope to show, ought not in strictness to be classed 
with them, as they rest on an entirely different basis, 

§ 3. The origin of such confusion is easy of 
explanation; it arises, I think, from the habit of 
laying undue stress upon the subjective side of Pro- 
bability, upon that which treats of the quantity of our 
belief upon different subjects and the variations of 
which that quantity is susceptible. It was seen that 
this vaiiation of belief is at most but an invariable ac- 
companiment of what is really essential to Probability, 
and is moreover common to other subjects as well. 
^y defining the science therefore from this side these 
other subjects would claim admittance into it; some 
of these, as Induction, have been accepted, but others 
have been somewhat arbitrarily rejected. Our belief 
in a wider proposition gained by Induction is, prior 

10^2 



148 XOGIC OP CHANCE. [CHAP. VI. 

to verificatioD, not so strong as that of the narrower 
generalization from which it is inferred. This being 
observed, a so-called rule of probability has been given 
by which it is supposed that this diminution of assent 
could in many instances be calculated. 

§ 4. But time also works changes in our conyiction ; 
our belief in the happening of almost every event, if 
we recur to it long afterwards, when the evidence has 
£ided from the mind, is less strong than it was at the 
time. Why are not rules of oblivion inserted in 
treatises upon Probability? If a man is told how 
firmly he ought to expect the tide to rise again, 
because it has already risen ten times, might he not 
also ask for a rule which should tell him how firm 
should be his belief of an event which rests upon a 
ten years' recollection 1 The infractions of a rule of 
this latter kind could scarcely be more numerous and 
extensive, as we shall see presently, than those of the 
former confessedly are. The fact is that the agencies, 
by which the strength of our conviction is modified, 
are so infinitely numerous that they cannot all be 
assembled into one science ; for purposes of definition 
therefore the quantity of belief had better be omitted 
from consideration, and the science defined from the 
other or statistical side of the subject, in which, as 
has been shown, a clear boundary line can be traced. 

§ 5. Induction, however, from its- importance 
does merit a separate discussion; a single example 



1 



SECT. 6.] LOOIO OP CHANCE. 149 

will show its bearing upon this part of our subject 
We are considering the prospect of a given man, A, A, 
living another year, and we find that nine out of ten 
men of his age do survive. In forming an opinion 
about his surviving, however, we shall find that there 
are in reality two very distinct causes which modify 
the strength of our conviction ; distinct, but in practice 
so intimately connected that we are very apt to over- 
look one, and attribute the effect entirely to the other. 

§ 6. (I) There is that which strictly belongs to Pro- 
bability; that which (as was explained in Chap, in.) 
measures our belief of the individual proposition as 
deduced from the general. Granted that nine men 
out often of the kind to which A. B, belongs do live 
another year, it obviously does not follow that he will. 
We describe this state of things by saying, that our 
belief of his surviving is diminished from certainty in 
the ratio of 10 to 9, or, in other words, is measured by 
the fraction ^. 

(II) But are we certain that nine men out of 
ten like him will live another year? We know that 
they have in time past, but will they continue to do 
so % Since A, B, is still alive it is plain that this pro- 
position is to a certain extent assumed, or rather ob- 
tained by Induction. We cannot however be as certain 
of the inductive inference as we are of the data from 
which it was inferred. Here, therefore, is a second 
cause which tends to diminish our belief; in practice 



150 LOOIC OF CHANCE. [CHAP. VI* 

these two causes always accompany each other, but 
in thought they can be separated. 

§ 7. The two distinct causes described above are 
very liable to be confused together, and the class df 
cases irom which examples are generally drawn 
increases this liability. The step from the statement 
*all men have died in a certain proportion' to the 
inference Hhey will continue to die in that proportion' 
is so slight a step that it is unnoticed, and the 
diminution of conviction that should accompany it is 
unsuspected. In what are called ^ priori examples 
the step is still slighter. "We feel so certain about the 
permanence of the laws of mechanics, that few would 
think it to be an inference when they believe that a 
die will in the long run turn up all its faces equally 
often, because other dice have done so in time past. 

§ 8, It has been already pointed out (in Chapter 
HI.) that, BO far as regards the definition of Probability 
as the science which discusses the modification of our 
belief, the question at issue seems to be simply this. 
Are the causes alluded to in (II) capable of being 
reduced to one simple coherent scheme, so that any 
universal rules for the modification of assent can be 
obtained from them? If they are, strong grounds 
will have been shown for classing them with (I), in 
other words for considering them as rules of pro- 
bability. Even then they might be rules of a different 
kind, contingent instead of necessary, but this objection 



1 



SECT. 8.] LOGIC OF CHAJfCE. 151 

miglit perhaps be overruled by the greater simplicity 
secured by classing them together. This view is, 
with various modifications, almost universally adopted 
by writers on Probability. Or, on the other hand, 
must these causes 'be regarded as a vast system, one 
might almost say a chaos, of perfectly distinct agencies; 
which may indeed be classified and arranged to some 
extent, but from which we can never hope to obtain 
any rules of wide generality which shall not be subject 
to constant exception? If so, but one course is left; 
to exclude them all alike from Probability. In other 
words, we must assume the general proposition, that 
which has been described throughout as our starting- 
point, to be given to us; it may be obtained by any 
of the numerous rules furnished by Induction, or it 
may be inferred deductively, or given by our own 
observation; its value may be diminished by its 
depending upon the testimony of witnesses, or being 
recalled by our own memory. Its real value may be 
influenced by these causes or any combinations of 
them; but all these are preliminary questions with 
which we have nothing directly to do. We assume 
our statistical proposition to be true, neglecting the 
diminution of its value by the process of attainment; 
we take it up first at this point and then apply our 
rules to it* We receive it in fact, if one may use the 
expression, ready-made, and ask no questions about 
the process or completeness of its manufkcture. 



152 XOGIC OP CHANCB. [cHAP. VI, 

§ 9. It is not to be supposed, of course, that any 
writers have seriously attempted to reduce to one 
system all the causes mentioned above, and to embrace 
in ODe formula the diminution of certainty to which 
the inclusion of them subjects us. * But on the other 
hand, they have been unwilling to restrain themselves 
from all appeal to them. From the first study of the 
science attempts have been made to proceed by the 
Calculus of Probability from the observed cases to 
adjacent and similar cases. In practice, as I have 
already said, it is not possible to avoid some extension 
of this kind. But it should be observed, that in these 
instances the divergence from the strict ground of ex-' 
perience is not in reality recognized; we have, it is 
tryie, wandered somewhat from it, and so obtained a 
wider proposition than our data, and therefore one 
of less certainty. Still we assume the two to be 
equally certain, or rather omit all notice of the 
divergence &om consideration. It is assumed that 
the unexamined instances will resemble the examined; 
the theory of the calculation rests upon the supposition 
that there will be no difference between them, and 
the practical error is insignificant simply because this 
differeuce is small. 

§ 10. But the rule we are now about to discuss, 
and which may be called the Rule of Succession, is of 
a very different kind. It not only recognizes the 
&ct that we are leaving the ground of past experience. 



} 



SECT. 12.] LOGIC OP CHANCE. 153 

but takes the consequences of this divergence as the 
express subject of its calculation. It professes to give 
a general rule, of unlimited application, for the measure 
of expectation that we should have of the reappearance 
of a phenomenon that has been already observed any 
number of times. This rule is generally stated somewhat 
as follows: "To find the chance of the recurrence of 
an event already observed, divide the number of times 
the event has been observed, increased by one, by the 
same number increased by two." 

§ 11. It must be confessed that this rule has 
been received in a thankless spirit. For considering 
what a number of events there are in the world, and 
how many are the ways in which they may happen, 
there is certainly no reason to fear that there will long 
be any want of occasions on which to appeal to the rule. 
The truth of it does not seem to be doubted by any of 
the writers on Probability; whilst those who have 
obtained their results from the mathematicians, as 
Archbishop Thomson, in his Law8 of Thoughty seem to 
regard it as standing on precisely the same footing as 
any of the other rules of the Science. It is our task 
at present to examine its claims to acceptance. 

§ 12. "We will begin with a detailed criticism of 
this rule. This is necessary both from the general 
acceptance it has received, and from the eminence of 
many of its supporters. Moreover, however much the 
rule itself may be found to fail when examined, many 



\ 



164 LOGIC OP CHANCE. [CHAP. TL 

principles of inference of real intrinsic importance 
will, I hope, be elicited in the course of our investiga- 
tion. We will afterwards shift the enquiry on to 
somewhat broader grounds, and examine some of the 
problems which have to be met in the case of any 
attempt to lay down rules of proof and discovery. 
These enquiiies do not very strictly belong to 
Probability, but they are so constantly encountered 
there that it seems essential to clear our way towards 
forming a decided opinion upon them. 

§ 13. Now there is one view of the question 
which deserves a passing notice, but nothing more. 
We may presume that the eminent writers who have 
accepted this rule do not regard it as the expression 
of a mere brute instinct. It is conceivable that on 
Physiological or Psychological grounds, the mere 
repetition of an event a certain number of times should 
excite a growing expectation of its recurrence. Em- 
ploying an illustration that shall at least have some 
connection with the derivation of the word, our 
impressions might be like those produced by hitting 
a soft plank with a mallet; each successive blow- 
deepens the impression. This would be to regard the 
rule as merely a mental law or instinct. But if it is 
to be considered as supplying real inferences about 
things, we cannot rest here. An instinctive belief 
may need to be corrected, to bring it into accordance 
with experience; and if not, it must at least submit to 



SECT. 14.] LOGIC OP CHANCE. 155 

justify itself by experience. It has been repeatedly 
stated already that to tell a rational being that his 
expectation of an event should be, say, f , can mean 
nothing else at bottom than this; — that events of the 
kind contemplated do really happen in that way three 
times out of four. 

§ 14. Is the rule then really true? Let us appeal 
to experience. In order that there shall be no unfiiir- 
ness we will begin with one or two examples selected 
by some of the most eminent writers on the subject. 
<juetelet informs us, that the man who has seen the 
tide rise ten days successively is right in entertaining 
an expectation of \l that it will do so again. Laplace 
has ascertained that, at the date of the publication of 
his work, one might have safely betted 1826214 to 1 
in favour of the sun's rising again. Since then how- 
ever time has justified us in laying longei* odds. De 
Morgan says, that a man who standing on the bank of 
a river has seen ten ships pass by with flags, should 
judge it to be 11 to 1 that the next ship will also 
carry a flag. Let us add an example or two more of 
our own. I have observed it rain three times suc- 
cessively, — I have found on three separate occasions 
that to give my fowls strychnine has caused their 
death, — ^I have given a false alarm of fire on three 
diflferent occasions and found the people come to help 
me each time. In each of these cases, then, T am to 
form an opinion of just the intensity of f in favour of 



156 LOGIC OP CHANCE. [cHAP. VI. 

a repetition of the phenomenon under similar cir- 
cumstances. But no one, we may presume, will assert 
that in any one of these cases the opinion so formed 
would be correct. In some of them our expectation 
would have been overrated, in some immensely under- 
rated. By calling the expectation wrong, it is not 
merely meant that it is frustrated in the particular 
case in question; this kind of failure, of course, is to 
be looked for in questions of Probability. But it is 
wrong in the long run. No amount of repetition in. 
our appeals to the rule in similar cases would lead us 
to even an average truth; this latter kind of truth we 
have a right to look for. With one single exception, 
and that a very doubtful one (the case of games of 
chance, bags and balls, &c.), the same objection could 
probably be brought against every possible application 
of this rule. Now granting that a formula of this 
kind, being given to any one, he might be justified in 
making use of it for a time, surely as soon as he has 
tried it and repeatedly found it lead him astray, it 
becomes his duty to reject and denounce it for the 
futura If, in the above examples, a person has really 
proportioned his expectation in the manner described, 
and has afterwards discovered by the examination of 
other instances of the same class, as he could hardly 
£Eiil to do, that his opinion had been grossly wrong, 
is he still to adhere to the rule for the future ? We 
need not blame him for doing what he did at the time; 



SECT. 16.] LOGIC OP CHANCE. 157 

he might have known no better; but is he to let the 
rule be published to the world as a ti'ue one? 

§ 15. It is merely evading the difficulty to assert, 
as is sometimes done, that the rule is to be employed 
in those cases only in which we do not know anything 
beforehand about the mode and frequency of occur- 
rence of the events. The truth or falsity of the rule 
surely cannot be in any way dependent upon the 
ignorance of the man who uses it. His ignorance 
affects himself only, and corresponds to no distinction 
in the things. In reality the two classes, viz. of cases 
in which we have and have not some preliminary 
information, are for the most part identical ; not, of 
course, identical to the same person at the same time, 
but in the sense that what one person does not know 
at present, he may hereafter, and others do know now. 
To say therefore that the rule refers to cases where 
there is no such preliminary information, is irrele- 
vant when the question is as to the correctness of the 
rule. We cannot fling the rule amongst mankind 
with the prescription attached that it is merely to be 
taken by the ignorant. They might have been in- 
clined to accept it once, but as soon as they know 
that its truth is denied by the better-informed, this 
amount of knowledge, though they possess no more, 
will be quite sufficient to prevent them from trusting 
to the rule. 

§ 16. I have said that the truth of the Eule of 



158 LOGIC OP CHANCE. [CHAP. VI* 

Suocession seems never to be doubted by mathema- 
tical writers on the subject. From this statement 
however exceptions must be made. Prof. De Morgan, 
is far too acute and philosophical a writer to accept 
the mle with the blind confidence with which it has 
sometimes been received. He regards it as famishing 
a miniimi/m value for the amount of our expectation. 

He would appear therefore to recognize only the 
instances in which our belief in the uniformity of 
nature, and in the existence of special laws of cmis- 
ation, comes in to aid that which we should enter- 
tain from the mere frequency of past occmTence of the 
particular event in questiou. His opinion is one from 
which I would dissent with deference, but it certainly 
appears to me to be irreconcilable with some of the 
instances given above. We have seen that there are 
cases in which the fact of a thing having already 
happened several times is a strong reason against its 
happening again. Can any marks be given by which 
these particular cases should be detected beforehand % 
and, if not, how can we assign a minimum value to 
the formula? A false alarm given several times in 
suocession is no imfair specimen of a considerable class 
of recurrences (others will be given in Chap. xu). 
Whilst such cases exist I cannot see that the rule can 
be regarded as correct. 

§ 17. It will not save the credit of the rule, in 
the above instance, to attempt to find its justification 



1 



SECT. 17.] LOGIC OF CHANCE. 159 

in some broader generalization, which controls and 
supersedes the narrower; to say, for example, that 
the measure of our expectation that the event will 
not recur is assigned by the number of times in which 
it has thus ceased to recur at that point before. This 
is shifting the ground, and instead of proving the 
correctness of the rule in question, offering to prove 
that of some other rule instead. It is like saying, 
in justification of some law which is accused of being 
pernicious, that the constitution in accordance with 
which this law was framed is itself on the whole 
highly advantageous. If we were speaking of the 
broader generalization, such a remark would be to 
the point, but we are not. 

The rule of succession informs us that, when an 
event has happened in a certain way four times, it is 
5 to 1 (some say at least 5 to 1) that it happens so 
next time. Against this we may adduce, not merely 
single cases, but whole classes of cases, in which such 
an opinion would be grossly erroneous. Each of 
these, of course, is a succession, and therefore has 
quite as much claim as any other succession to be 
included within the rule ; it does not save the credit 
of the rule to say that it applies to another and quite 
different succession. How is any one to know before- 
hand whether it applies to his own particular cir- 
cumstances or not, unless it be, as of course it pro- 
fesses to be, perfectly general ? 



160 LOGIC OP CHANCE. [CHAP. VI. 

§ 18. "When we look at the above more closely 
we shall find, I think, that it is really a defence of the 
rule, if defence it can be called, which, at the absolute 
sacrifice of its validity as a rule strictly so teimed, 
would seek to retain and justify it as a general prin^ 
ciple. Admitting it to be true that the rule fails, 
if we infer, from the fact of its having rained three 
days running, that it will rain again, what else in 
fact are we doing, it may be said, but abandoning 
the rule in one form to retain it in another ] If we 
experience another such succession of three rainy 
days, we now do not expect it to be followed by a 
fourth of the same kind; in so doing, it may be urged, 
are we not necessarily resorting to the very rule that 
we professed to discard? are we not now making a 
precisely similar succession, not indeed of individual 
rainy days, but of siLCcesaiona of rainy days, and so 
forming our anticipation] 

§ 19. Whatever might be said for the above de- 
fence, it is fatal, as I have already remarked, to the 
integrity and utility of the rule, as a rule. To follow 
up such an enquiry as that to which it seems to con- 
duct us, would be to wander far from the province 
of Probability. It would lead to an investigation, 
not exactly into the direct formation of rules of in- 
ference, and certainly not into their correctness, but 
into the ultimate principle or constitution of the 
mind upon which all Induction is founded. It would 



1 



SECT. 20.] LOGIC OF CHANCE. 161 

involve, as it appears to me, a discussion of the fun- 
damental laws of association, upon which all infer- 
ences about things might be conceived ultimately to 
rest; a discussion which would belong more pro- 
perly to Psychology than to any branch of Logic. 

§ 20. It is quite true that there must be some 
mental link to bind together the examined and the 
unexamined cases, before we can make any new iur 
ferences about the latter. Without such a link there 
could of course be no extension beyond the strict 
limits of past experience. It is also obvious that a 
stronger degree of conviction or anticipation is pro- 
duced in some instances than in others, the strength 
of this conviction depending unquestionably, in part, 
upon the degree of resemblance between the examined 
and the unexamined cases; it may also depend in part 
upon the number of times in which the examined 
cases have been already observed. But it need not 
be the case, as seems to be commonly supposed, that 
the strength of conviction must increase uniformly 
from zero towards certainty in proportion to the 
number of these observed instances of recurrence. It 
is at least equally possible, as is held by some writers, 
that the conviction should exist in its full degree 
after the first occurrence of the event, viz. that our 
primitive impulse should be to fully believe that any 
two things which had been once observed together 
would so occur again; this belief becoming of course 

11 



162 LOGIC OP CHANCE. [CHAP. VI. 

altered in amouut and often removed bj subsequent 
experience. 

But whichever way the matter be settled it is not 
easy to see how such considerations can have any 
bearing upon rules of inference at the point at which 
they are taken up in Logic or Probability. The psy- 
chological principles just mentioned lie at an immense 
depth below the surface of these rules, and assume a 
very different form before they emerge into the shape 
of laws of inference for minds of mature intelligence. 
In this latter shape they must, of course, submit to 
be tested by experience, as we have tested them 
throughout; but I cannot see that we are concerned 
with the process of their growth, or the germ out of 
which they have been developed. 

I find it difficult to ascertain precisely from La- 
place's Essay what his view of this Rule of Succession 
is. On the one hand he certainly appeals to it as a 
valid rule of inference, but on the other hand he 
enters into decidedly psychological and even physio- 
logical explanations in the latter part of his Essay. 
But he does not appear to perceive the feet that by 
converting any such formula into an ultimate prin- 
ciple we do in reality abandon it as a practical rule. 

§ 21. But if this rule be regarded strictly as a 
rule, the reader may well be supposed to enquire, by 
this time, how it was ever discovered, and whence it 
obtains its proof? We have not far to seek for in- 



\ 



SECT. 21.] LOGIC OF CHANCE. 163 

formation upon these points. It certainly was not 
discovered from observation or experience of nature^ 
for this, ajs we have seen, contradicts it in almost 
every instance. Nor was it discovered by observation 
of the mind; for this only leading to a knowledge of 
what men do bdiieve, and not of what they should 
believe, can be no valid guide in drawing inferences 
about things without us. 

There seems to remain but one way. *We may 
discover amidst the infinite complexity of nature some 
class of objects that may be regarded as a fair type 
or sample of all the rest. The play of the different 
agencies at work elsewhere may be there laid bare to 
view, as it were, so that we may feel certain that so 
fBLP as regards the succession of phenomena we have 
arrived at some of the fundamental priaciples of the 
universe. It is obvious that the connection between 
this class of objects and the rest of nature must be of 
no transient or superficial character. But when we 
have discovered this connection we shall be able to 
infer a rule of such broad generalization that in no 
single instance will any man be able to act upon it. 
Such an example has been discovered by some of the 
supporters of this rule. What then are the data by 
which this grand generalization is drawn? by which, 
suocording to Laplace, we feel a confidence, as the sun 
sets, of more than a million to one that it will rise 
again? and by which each generation of husbandmen 

11—2 



164 LOGIC OP CHAKCE. [CHAP. VI. 

may go on sowing and reaping with a deeper per- 
suasion than their fathers possessed before them, that 
seed-time and harvest and summer and winter will 
not fail] A study of the works of these writers will 
discover that it is a bag containing balls of a black 
and white colour. Kules, of more or less accuracy, 
are established as to the surmises we may form about 
the proportion of different colours in the bag, after we 
have drawn a few, and therefore of the proportion that 
will continue to be given in future. The supposition 
apparently slips in somewhere about here that the 
universe is constructed on the same principle as such 
a bag, from which the rule, in all its generalization, is 
supposed at once to follow. 

The above is no caricature of the process by 
which this Bule of Succession is commonly obtained. 
If there are any persons who believe that something 
of greater value than formulae for the manipulation of 
symbols can be obtained in this way, they should mark 
the two following chasms in the logic, across neither 
of which is it easy to find a passage. The first lies 
between the premises and conclusion of the argument 
by which we infer from a limited number of drawings 
what was the number of balls of each colour in the bag. 
This will b3 referred to again in Chap. Vlii. The 
second lies in our way when we try to proceed from 
such a rule as this about balls to somewhat more 
general conclusions about the phenomena of natui*e. 



1 



SECT. 23.] XOGIC OP CHANCE. 165 

§ 22. It is worth pausing for a moment in order 
to understand the nature of the rule if it be supposed 
to tbe obtained in this way. It need not lead us to 
truth in any single case; this of course is not ex- 
pected in Probability. Nor need it lead us to truth 

in the average of any class of cases; this might fairly 

fl 

Jiave been expected. But the rule will profess that 
an appeal to it in all classes of cases whatever, by all 
mankind, would lead to truth, and that it is ready 
to submit to this test of universal and incessant 
ejxperience. On such a view the rule seems almost 
equally to evade attack and defence; its vagueness 
and generality are its protection. With equal reason 
might we attempt to take thp average size of all 
measurable things, and then determine to act upon 
the assumption that every thing is just of that size. 
We might conceive a sort of justification of the aver- 
age conduct of all mankind if they always acted so; 
but how would any one person prosper during a 
limited time on these terms ] Any rule of discovery 
that Inductive Logic can recognize must surely be 
specialized by the imposition of some limil^ of time 
and place to its applicability. 

§ 23. The Eule has been discussed, during this 
chapter, in its simplest form. Our criticisms however 
will apply with equal or greater force to the more 
complex form, in which it is attempted to determine 



166 LOGIC OP CHANCE. [CHAP. TI. 

the cliance, not of one more recurrenoe only, but of 
any number of recurrences. 

§ 24. So mucb then for this Rule of Succession. 
Not that we have yet exhausted its shortcomings; 
for, as we shall see presently, it does not merely 
mislead us by giving one determinate but incorrect 
answer; it perplexes us by the offer of several <fis- 
cordant and often contradictory answers, all of them 
incorrect. Nothing but the celebrity of its supporters, 
and the general acceptance it has met with, have 
been our reasons for examining it so minutely as we 
have done. 

But to simply criticise and reject it is not sufficient. 
We should like to know somewhat more fully why it 
fails so utterly, and whether anything can be sub- 
stituted for it, for the end it seeks, viz. the extension 
of our inference beyond the limits of direct observation, 
is one which is desirable and necessary if we are ever 
to obtain information about things in general. The 
rule, as has been already said, seems to involve con- 
siderable concision between Probability and Induction. 
This confiision can only be resolved, and the portion 
of truth mixed up in it elicited, by trying back some 
steps, and commencing with an analysis of the province 
and nature of Induction. This is a process which will 
be entered on in the next chapter. 



CHAPTER VII. 

INDUCTION, AND ITS CONNECTION WITH 
PEOBABILITY. 

§ 1. A RULE was examined at some length in 
the last chapter, the object of which rule was to 
enable ns to make inferences about instances as yet 
unexamined. It was professedly, therefore, a rule of 
an inductire character. But, in the form in which 
it is commonly expressed, it was foimd to fail utterly, 
proving, when applied to the phenomena of nature, 
to be generally at least false or inapplicable. It is 
reasonable therefore to enquire at this point whether 
Probability is entirely a formal or deductive science, 
or whether, on the other hand, we are able, by means 
of it^ to make valid inferences about instances as yet 
unexamined. This question has been already in part 
answered at the commencement of the last chapter. 
I propose in the present chapter to give a fuller 
investigation to this subject, and to describe, as mi- 
nutely as limits will allow, the nature of the Con- 
nection between Probability and Induction. We shall 
find it advisable for clearness of conception to com- 
mence our enquiry at a somewhat early stage. We 
will travd over the ground however as rapidly as 



168 LOGIC OP CHANCE. [CHAP. VII. 

possible until we approach the boundary of what can 
properly be termed Probability. 

§ 2. Let us then conceive some one setting to 
work to investigate nature, under its broadest aspect, 
with the view of systematizing the facts of experience 
that are known, and thence discovering others which 
are at present unknown. He obsei-ves a multitude 
of phenomena, physical and mental, contemporary and 
successive. He enquires what connections are there 
between them ] what rules can be found, so that some 
of these things being observed I can infer others from 
them ? We suppose him, let it be observed, delibe- 
rately to investigate the things themselves, and not 
to be turned aside by any prior enquiry as to there 
being laws under which the mind is compelled to 
judge of the things. This may arise either from a 
disbelief in the existence of these mental laws, and 
a consequent conviction that the mind is perfectly 
competent to observe and believe anything that ex- 
perience offers, and should believe nothing else, or 
simply from a preference for investigations of the 
latter kind. In other words, we suppose him to reject 
Formal Logic, and apply himself to a study of objec- 
tive existences. 

§ 3* His task at first might be conceived to be a 
slow and tedious one. It would consist of a gradual 
accumulation of individual instances, as marked out 
and connected together by resemblances ; these would 



I 



SECT. 4.] LOGIC OP CHANCE. 169 

Jiben be summed up in general propositions, fi-om 
wbicb inferences could afterwards be drawn. These 
inferences could, of course, contain no new facts, they 
would only be repetitions of what he or others had 
previously observed. The principles of ordinary logic 
•would of course be needed now, but these would 
rather be regarded as being determined by the con- 
stitution of the things than by that of the mind 
in observing the things. So far we have supposed 
the observer not to have advanced beyond the province 
of applied logic in its usual sense. 

§ 4. But a very short course of observation would 
suggest the possibility of a wide extension of his 
information. Experience itself would soon detect that 
events were connected together in a regular way ; he 
would ascertain that there are *laws of nature.' 
Coming with no a priori necessity of believing in 
them, he would soon find that as a matter of fact they 
do exist, though he could not feel any certainty as to 
the extent of their prevalence. The discovery of this 
arrangement in nature would at once alter the plan of 
his proceedings. His main work now would be to 
find out by what means he could best discover these 
laws of nature. 

An illustration may assist. Suppose I were en- 
gaged in breaking up a vast piece of rock, say slate, 
into small pieces. I should begin by wearily working 
through it inch by inch. But I should soon find the 



170 LOGIC OP CHANCJB, [CHAP. Vn. 

prooess completely changed owing to the ezistenoe of 
cleavage. By this arraDgeoient of things a very few 
\Agws would do the work, — not^ as I had at first 
sapposed, to the extent of a few inches, — but right 
through the whole mass. In other words, by the pro* 
cess itself of cutting, as shewn in experience, and by 
nothing else, a constitution would be detected in the 
things that would make that process vastly more 
easy and extensive. Such a discovery would of course 
change our tactics. Our principle object would thenoe- 
forth be to ascertain the extent and direction of this 
cleavage. 

Something resembling this is found in Inducti<Hi. 
The discovery of laws of nature enables the mind 
to dart with its inferences from a few faasta completely 
through a whole class of objects, and thus to acquire 
results the successive individual attainment of which 
would have involved long and wearisome investiga- 
tion. We have no demonstrative proof that this state 
of things is universal ; but having foimd it prevail 
extensively, we go on with the resolution at least 
to try for it everywhere else, and we are not disap^ 
pointed. From propositions obtained in this way, 
or rather from the origina] facts on which these pro- 
positions rest, we can make new inferences, not indeed 
with absolute certainty, but with a degree of convic- 
tion that is of the utmost practical use. We have 
gained the great step of being able to make trust- 



SECT. 6.] LOGIC OF CHANCR. 171 

Worthy generalizations. We oonclude^ for instance, 
not merely that John and Heniy die, but that all 
xncm die. 

§ 5. The above investigation contains, I think, 
a tolerably correct outline of the nature of the In- 
dactive inference, as it presents itself in Material or 
Phenomenalist Logic*. It involves the distinction 
drawn by Mr Mill, and with which the reader of his 
System of Logic will be familiar, between an inference 
drawn ttcoording to a formula and one drawn from 
a formula. We do in reality make our inference from 
the data afforded by experience directly to the con- 
clusion ; it is a mere arrangement of convenience to 
do so by passing through the generalization. But it 
is one of such extreme convenience, and one so neces- 
sarily forced upon us when we are appealing to our 
own past experience or to that of others for the 
grounds of our conclusion, that practically we find it 
the best plan to divide the process of inference into 
two parts. The first part is concerned with establish- 
ing the generalization; the second (which contains 
the rules of ordinary logic) determines what condii* 
sions can be drawn from this generalization. 

§ 6. We may now see our way to ascertaining 
the province of Probability and its relation to kindred 
sciences. Inductive Logic gives rules for discovering 

• I have borrowed this latter term from Professor Grote^s 
admirable and suggestive Exploratio PhUosophica. 



172 LOGIC OF CHANCE. [cHAP. VII, 

such generalizations as those spoken of above, and for 
testing their correctness. If they contain universal 
propositions it is the part of ordinary logic to de* 
termine what inferences can be made from and by 
them ; if, on the other hand, they contain proportional 
propositions, that is, propositions of the kind described 
in our first chapter, they are handed over to Proba- 
bility. We find, for example, that three infants out 
of ten die in their first year. It belongs to Induction 
to say whether or not we are justified in generalizing 
our observation into the assertion. All infants die in 
that proportion. When such a proposition is obtained, 
whatever may be the value to be asSgned to it, we 
recognize in it a series of a familiar kind, and it is 
at once claimed by Probability. 

In this case the division into two parts, the induc- 
tive and the ratiocinative, seems something more than 
one of convenience; it is imperatively necessary for 
clearness of thought and arrangement. It is true 
that in almost every example that can be selected we 
shall find both of the above elements existing together 
and combining to modify the degree of our conviction, 
but when we come to examine them closely it appears 
to me that the grounds of their cogency, the kind of 
conviction they produce, and consequently the rules 
which they give rise to, are so entirely distinct that 
they cannot possibly be harmonized into a single con- 
sistent system. 



1 



SECT. 7.] LOGIC OF CHANCE. 173 

The common opinion therefore which regards In- 
ductive formulse as composing a portion of Probability, 
and which finds utterance in the Kule of Succession 
criticised in our last chapter, cannot, I think, be 
maintained. It would be more correct to say, as 
stated above, that Induction is quite distinct from 
Probability, but yet co-operates with almost all its 
inferences. By the former we determine, for exr 
ample, whether we can safely generalize the proposir 
tion that four men iu ten live to be fifty ; supposing 
such a proposition to be generalized, we hand it over 
to Probability to say what sort of inferences can be 
deduced from it 

§ 7, We may now see clearly the reasons for the 
limits within which causation* is necessarily required, 
but beyond which it is not needed. To generalize a 
formula so as to make it extend from the known to. 
the unknown, it is clearly essential that there should 
be a certain permaaence in the order of nature; this 
permanence is one form of what we mean by causa- 
tion. If the circumstances under which men live and 
die remaining the same, we could not infer that four 
men out of ten would continue to live to fifty, because 
in the case of those whom we had observed this pro- 
portion had hitherto done so, it is clear that we should 
be admitting that the same antecedents need not be 

* A separate chapter will be devoted to the discussion of 
Causation, 



174 LOGIC OP CHANCaSL [chap. VII. 

followed by the same consequents. This uDiformity 
being what the Law of Causation asserts, the truth of 
the law is clearly necessary to enable us to obtain our 
generalizations; in other words, it is necessary for the 
Inductive part of the process. But it is equally dear, 
I think, that causation is not necessary for that part 
of the process which belongs to Probability. Pro- 
vided only that the truth of our generalisations is 
secured to us, in the way just mentioned, what does it 
matter to us whether or not the individual members 
are subject to causation? For it is not in realily 
about these individuals that we make inferences. As 
this subject is more fully treated in another chapter, 
I need not make any farther allusion to it here. 

§ 8. The above description of the process <^ ob- 
taining these generalizations must suffice for the pre- 
sent. Let us now turn and consider the means by 
which we are practically to make use of them when 
they are obtained. The point which we had reached 
in the course of the investigations entered into in the 
fourth chapter was this: — Given a series of » certain 
kind, we could draw inferences about the members 
which composed it; inferences, that is, of a particular 
kind, the value and meaning of which were fuUy dis* 
cuBsed in their proper place. 

f 9. We must now shift our point of view a little ; 
instead of starting, as in the third chapter, with a 
determinate series supposed to be given to us, let us 



1 



SECT. 9.] IjOGIO OF CHANCE. 175 

afisiune that the mdividual only is given, and that the 
work is imposed upon us of finding out the appropriate 
series. How are we to set about the task ? Before 
our data were of this kind : — !Kight out of ten men, 
aged fifty, will live ten years more, and we ascer* 
tained in what sense, and with what certainty, we 
eould infer that, say, John Smith, aged ^£tj, would 
live ten years. 

Let us now suppose, instead, that John Smith 
presents himself, how should we in this case set about 
obtaining a series ] In other words, how should we 
collect the appropriate statistics 1 It should be borne 
in mind that when we are attempting to make real 
inferences about things as yet unknown, it is in this 
form that the problem will practically present itself. 

The answer to this question may seem to be ob- 
tained by a very simple process, viz. by counting how 
many men of the age of John Smith, respectively do 
and do not live for ten years. In reality however the 
process is far fironi being so easy as it appears. For it 
must be remembered that each individual has not one 
distinct and appropriate series, to which, and to which 
alone, it properly belongs. We may indeed be prac- 
tically in -the habit of considering it under such a 
single aspect, and it may therefore seem to us more 
^miliar when it occupies a place in one series rather 
than in another, but such a practice is merely cus- 
tomary on our part, not .essentiaL It is obvious that 



176 LOGIC OF CHAKCE. [cHAP. VH. 

every individual thing or event has an indefinite num- 
ber of properties or attributes observable in it, and 
might therefore be considered as belonging to an inde- 
finite number of difibrent classes of things. By be- 
longing to any one class it of course becomes at the 
same time a member of all the higher classes, the 
genera, of which that class was a species. But, more- 
over, by virtue of each attribute which it possesses, it 
becomes a member of a class not necessarily conter- 
minous with any of the other classes. John Smith is 
a consumptive man say, and a native of a northern 
climate. Being a man he is of course included in the 
class of vertebrates, also in that of animals, as well as 
in any higher classes that there may be. The pro- 
perty of being consumptive refers him to another class, 
narrower than any of the above; whilst that of being 
bom in a northern cUinate refers him to a totally dis- 
tinct class, not conterminous with any of the rest, for 
there are things bom in the north which ai'e not men. 
Now when he stands before us as an individual, it is 
altogether arbitrary under which of these aspects we 
view him, whilst the conveyance of our opinion to 
others depends on the name J)y which we call him. I 
do not mean of course that we need know of all these 
properties so as to be acquainted with the correspond- 
ing classes; it is quite sufficient that in any ordinary 
state of knowledge and intelligence we are familiar 
with some such properties, and recognize in conse- 



1 



ilBCT. 10.] LOGIC OP CHAirClC 177 

qtieboe that the individual maj be referred t6 several 
different classes. To the student of logic all this will 
be too fiimiliar to. need explanation. 

This variety of classes to which the individual may 
be referred, owing to its possession of a multiplicity of 
aittributes, has an important bearing oH the procesft of 
inference which was touched upon in the earlier sec* 
tions of this chapter, and which we are now about to 
examine more in detail. . - * > 

§ 10. It will serve to bring out more clearly 

some of the peculiarities in the case of Probabili4iy9 

•of the step which we are now about to take, if we 

iirst examine the form it assumes in the case of (Or< 

dinary Logic. Suppose, then, that I wish, to ascertain 

.whether a certain man, who is amongst other .things 

>a consumptive pauper, aged thirty, will live tweaiy 

years more. The terms in which the man .is,, thus 

, introduced refer him to different classes, in the way 

Just described. Corresponding to these classes therip 

will probably be ft number of propositions discovered 

by previous observations or Inductions They may 

be«uch as these following ; — Some men live to sixty. 

No consumptives live to forty. No pauper lives to 

. fifty. At the stage of enquiry which we at present 

oooupy we may of course suppose any number of i^uch 

propositiom^ that we may need to be ready at hapd. 

It need hardly be said that those given here do. ifpt 

profess, to be true, but are only chosen for illiiistrat^pn. 

12 



178 LOOIO OF CHANGE. [cHAP. VII. 

Erom the first of these propositions we can infer 
nothing to our purposa By either the second or the 
third we can answer our enquiry. To the logical 
reader it need not be pointed out that the process 
now under consideration is that of finding middle 
terms which shall embrace the individual in question. 

Connected with the above logical process there are 
t«o considerations to which the reader's attention is 
especially directed; Firstly, if we can infer anything 
at all, we do so absolutely; assuming our premises 
to be true, we either know our residt for certain, 
or know nothing about it. Secondly, no one of the 
above possible major propositions can ever contradict 
the others, or be to any extent at variance with 
them. To suppose this possible would be in e&ct 
to make two contradictory assumptions about matters 
of &ct 

§ 11, Biit now observe the difference when we 
attempt to take the corresponding step in Probability. 
For the question stated above, substitute the follow- 
ing : — Will the man in question live one year 1 We 
. shall find no universal propositions here, but we may 
find an abundance of proportional propositions. .They 
will be of the following description (the data are 
purely imaginary) : — Of men aged thirty, ninety-nine 
in a hundred live to thirty-one ; of paupers of that 
age, nineteen in twenty survive one year; of con- 
sumptive men, but one in three survive. In both 



1 



myr. 12.] logic of chance. 179 

of the respects to which attention was drawn, propo- 
sitions of this kind offer a marked contrast to those 
last considered; they do not assert unequivocally 
yes or no, but give what in the case of the individual 
is a kind of qualified or hesitating answer. And 
these answers, though they cannot formally contra* 
diet one another, may yet be more or less in con- 
flict. 

§ 12. Hence it is obvious that in the attempt 
to draw a conclusion we may be placed in a position 
of some perplexity, but it is a perplexity that may 
present itself in two forms, a mild and an aggravated 
form. 

The mild form occurs when the different classes 
deferred to above are successively included one within 
another, for here our sets of statistics, though dif- 
ferent, will be found scarcely, if at all, at variance 
with one another. The only difference between one 
Bet and another is that as we ascend in the scale to 
the larger genera the statistics become less appropriate, 
and the information they afford, therefore, less ex- 
plicit and accurate. Let us examine this case first. 
What we originally wanted to ascertain, be it re- 
membered, is whether John Smith will die within 
one year. But all knowledge of this fact being un- 
attainable,^ we felt justified (under the restrictions 
mentioned in Chapter in.) in substituting as the 
best available equivalent for such information, the 

12— a 






180 LOGIC OF OHANCB. [CHAP. tin 

following statisticdl enquiry, What proportion of mea 
in his circumstances die? 

But then there arises some doubt and ambiguity 
as to what exactly is to be understood by his circum- 
stances. Knowing well what they are in themselves 
we are in perplexity as to how many we ought to take 
into account. We might confine our attention to 
those only which he has in common with all animals; 
if so we should have such statistics as thi^^ ninety- 
nine animals out of a hundred will die within a year. 
But we reject this and prefer a narrower series, for 
the obvious reason that by so doing we secure our 
being more often right*, and, when we are not ijght, 
of being less flagrantly in error. 

The above reasons ai-e conclusive against taking 
too wide a class, but how can our class be too narrow? 
John Smith is not only an Englishman, he may be 
also from Suflblk, a farmer, &c. Why do we also 
reject these narrower classes? We do reject them, 
but it is for what may be called a practical rather 
than a theoretical reason. It must be borne in mind 
(as was shown and illustrated in the first chapter), 

* More often right, that is, when we make a succession of 
such judgments about men. Some predetermination, not neces* 
sarily as to the particular class, but at least as to the sort of 
class to which we mean to direct our attention, cannot be avoided. 
In other words, we must assume the existence not only of a sys- 
tem of classification, but of certain channels in which our judg- 
ments about the objects included in these classes principally lie. 



\ 



^ECT. 13.] IrOGIC OF CHANOEt J81 

that it is essential to the sort of series we want that 
it should contain a considerable number of terms^ 
Kow many of the attributes of any individual are so 
rare that to take them into account would, be at 
variance with this fundamental position of our science^ 
that we are properly only concerned with the averages 
of large numbers. The more special the statistics 
the better, if we could only get enough of them, and 
80 make up the requisite large numbers, but this ia 
unfoi'tunately impossible. We are therefore obliged 
to neglect one attribute afber another, at the probable^ 
risk of increased inaccuracy, for at each step of thist 
kind we diverge more and more from the sort of 
instances that we really want. We make our stand 
finally at the point where we can first obtain sta-r 
tistics drav^n from a sufficiently large range of obseiw 
Tation. 

§ 13* In such an example as the one mentioned 
above, where one of the classes — ^man — ^is a natural 
kind, there is such a complete break at this point, 
that on the one hand no one would ever think of 
introducing any refei-ence to the higher classes with 
fewer attributes, such as animal or organised being. 
And on the other hand the inferior classes, such aa 
&xmer or inhabitant of Suffolk, do not differ suffix 
ciently in their characteristics fi'om the class mem 
to make it worth our while to attend to themr 2Tow 
and then indeed these characteristics do become impose 



182 JJOQIO OF CHAKOB, [CHAP. VJl* 

tant| and whenever this is the case we concentrate 
our attention upon the class to which they correspond, 
that is, the class which is marked out by their pos- 
session. Thus the qualiiy of consumptiveness sepa- 
rates any person off from his fellow-men so widely 
that statistics about the lives of consumptive men 
would differ materially from those which refer to men 
in general And we see the result ; if a consumptive 
man can effect an insurance at all, he must do it 
for a much higher premium, calculated upon his spe> 
cial circumstances. In other words, the attribute is 
sufficiently important to mark off a fresh series. 

§ 14. Where the classes thus correspond to na- 
tural kinds the process is not difficult; there is almost 
always some one of these kinds which is so univer- 
sally recognized to be the appropriate one that most 
persons are quite unaware of there being any necessity 
£:>r a process of selection. Except in the oases where 
a man has a sickly constitution, or follows a dan- 
gerous employment, we never think of collecting sta-* 
tbtics for him from any class but that of men in. 
general of his age in the country. 

When, however, these successive classes are not 
ready marked out for us by nature, and thence ar- 
ranged in easily distinguishable groups, the process 
iB more obviously arbitrary. Suppose we were con- 
sidering the chance of a man's house being bomt 
down, with what collection of attributes should we- 



SECT. 15.] LOGIC OF (MA3XCSL 183^ 

rest content in this instance? Should we embrace^ 
all kinds of buildiogs, or only dirdiling-houses, or 
confine ourselves to those where there is much wood, 
or those which have stoves ? All these attributee^ 
and a multitude of others may be present^ and, if so, 
they are all circumstances which help to modify our 
judgment. We must be guided here by the statistics 
which we happen to be able to obtain in sufficient 
numbers. Bough distinctions of this kind are prac«v 
tically drawn in Insurance offices, by dividmg risks 
into ordinary, hazardous, and extrarhazardous. We 
examine our case, refer it to one or other of these 
classes, and then form our jud^ent upon its prospecta 
by the statistics appropriate to its class. 

So much for what I have called the mild form 
in which the ambiguity occurs j but there is an aggra-^ 
vated form in which it may show itself and which 
seems to place us in far greater perplexity. 

§ 15. Suppose that the different classes mentioned 
above are not included one within the other. We may 
then be quite at a loss which of the statistical tables to 
employ. Let us assume, for example, that nine out of 
ten Englishmen are injured by residence in Madeira^ 
but that nine out of ten consumptive persons are bene* 
fited by such a i-esidence. These statistics, though 
imaginary, are possible and perfectly compatible. 
John Smith is a consumptive Englishman ; are we to 
recommend a visit to Madeira in his ease or not i la 



184' LOGIC OF CHAKCB. [CHAP. Vlt. 

other words, what inferences are we to draw about the 
probability of his death ) Both of the statistical 
tables apply to his case, but they would lead us to^ 
directly cimtradictory conclusions. I do not mean, of 
course, contradictory precisely in the logical sense of 
that word, for one of these propositions does not assert 
that an event must happen and the other deny that it 
must; but contradictory in the sense that one would 
cause us in some considerable degree to believe 
what the other would cause us in some considerable 
degree to disbelieve. This refers, of course, to the 
individual events, the statistics are by supposition in 
no degree contradictory.. Without further data there- 
fore we can come to no dedsion. 

Practically, of course, we should make our choice 
by the consideration that the state of a man's lungs 
has probably more to do with his health than the 
place of his birth has; that is, we should conclude 
that the duration of life of consumptive Englishmen 
corresponds much more closely with that of consump- 
tive persons in general than with that of their healthy 
countrymen. But this is, of course, to import alien 
^nsiderations into the question. The data, as they 
are given to us, and if we confine ourselves to them, 
leave us in absolute perplexity. It may be that the 
consumptive Englishmen almost all die when trans- 
ported into the other climate; it may be that they 
libnQst all recover. If they die, this is in obvious 



8BCT. 16.] LOOIO OF CHANCE. 185 

accordance with the first set of statistics; it will be 
found in accordance with the second set through the^ 
&ct of the foreign consumptives profiting by the 
<^nge of climate in more than their due proportion,' 
A similar explanation will apply to the other altera 
native. The problem is therefore left absolutely inde- 
terminate, for we cannot here appeal to any general 
rule 80 simple and so obviously applicable as that 
which, in a former case, recommended us always to 
prefer the more special statistics, when sufficiently 
extensive, to those which are wider and more general 
We have no means here of knowing whether one is 
more special than the other; in fact such a term as 
special is inappropriate. 

§ 16. And in this no difficulty can be found, so 
long as we confine ourselves to a just view of the sub- 
ject. Let me again recall to the reader's mind what 
our present position is; we have substituted for know- 
ledge of the individual (finding that unattainable) a 
knowledge of what occurs in the average of similar 
cases. But the conception of similarity in the cases 
introduces us to a perplexity; we manage indeed to 
evade it in many instances, but here it is inevitably 
forced upon our notice. There are here two aspects 
of this similarity, and they introduce us to two dis^ 
tinot averages. Two assertions are made as to what 
happens in the long run, and both of these assertions, 
by supposition, are verified. Of their truth there 



186 LOGIC OF CHANGEr [cHAP. YII.} 

need be no doabt, for both were obtained from expe»'> 
rienoe. 

§ 17. It mstiy perhaps be supposed that such an 
example as this is a veductio ad ahsurdum of the prim*; 
ciple upon which Life and other Insurances are 
founded. But a moment's consideration will show: 
that this is quite a mistake, and that the principle of 
Assurance is just as applicable to examples of this 
kind as to any other. An Office need find no difficulty^ 
in the case supposed. They migh^ (for a reason to be 
nientioned presently, they probably wotdd not) insure 
the individuifcl without inconsistency at a rate deter- 
mined by either average. They might say to him» 
' You are an Englishman. O ut of the multitude of Eng*: 
lish who come to us nine in ten die if they go to Ma- 
deira. We will insui'e you at a rate assigned by these 
statistics, knowing that in the long run all will come 
right. You are also consumptive, it is true, and we 
do not know what proportion of the English are con* 
sumptive, nor what proportion of English consump- 
tives die in Madeira. But this does not really matter- 
for our purpose. The formula, nine in ten die, is ia 
reality calculated upon these unknown proportions; for^ 
being a statistical result, it must involve all such pro- 
portions. In other words, the unknown elements^ 
must, in regard to all the effects which they can pro- 
duce^ have been already taken into account. All 
the unknown conditiQus,^ therefore, will be found to 



1 



SECT. 18.] LOGIC OP CHANCEL 187 

lirmDge themselves in accordance with the above sta- 
tistical result And this is sufficient for our purpose. 
But precisely the same language might be held to him 
if he presented himself as a consumptive man; that is 
to say, the Office could safely carry on its proceedings 
upon either alternative. 

This would, of course, be a very imperfect state 
for the matter to be left in. The only rational j^aiE 
would be to isolate the case of consumptive English*' 
men, so as to make a separate calculation for their 
droumstances. This calculation would then at once 
supersede all other tables; for though, in the end, it 
Could not arrogate to itself any superiority over the 
otii^rs, it would in the mean time be marked by fewer 
and slighter aberrations from the truth. 

§ 18. The real reason why the Insurance office 
eould not long work on the above terms is of a very, 
different kind from that which some readers might con* 
template, and belongs to a class of considerations which, 
have been strangely neglected in the attempts to con-> 
struct sciences of the iiifierent branches of human con- 
duct. It is nothing else than that annoyiug contin- 
gency to which prophets since the time of Jonah have 
been subject^ of uttering suicidal prophecies; of pub- 
lishing conclusions which are perfectly certain when 
every condition and cause but one have been takeU' 
into account, that one being the effect of the pro-c 
pheoy itself upon those to whom it refers. 



188 LOGIC OP CHANCE, [CHAP. VII* 

In our example above, the Of&ce would get on very: 
well until the consumptiye persons found out what 
much better terms they could make by announcmg 
their consumptiveness, and paying the premium ap* 
propriate to that class. But if they did this they 
would of course be disturbing the statistics. The 
tables were based upon the assumption that a certain 
fixed proportion (it does not matter what proportion) 
of the English liyes would continue to be consump- 
tive lives, which, under the supposed circumstances^ 
would probably soon cease to be true. When it is 
said that nine Englishmen out of ten die in Madeira, 
it is meant that of those who come to the Office, as the 
phrase is, at random, or in their fair proportions, nine- 
tenths die. The consumptives are supposed to gO; 
there just like red-haired men, or poets, or any other 
special clajss. Or they might go in any proportions 
greater or less than those of other classes, so long as 
they adhered to the same practice throughout. The 
tables are then calculated on the continuance of this 
state of things; the practical contradiction is in sup- 
posing such a state of things to continue after, the 
people had once had a look at the tables. If we 
merely make the assumption that the publication of 
these tables made no such alteration in the conduct 
of those to whom it referred, no hitch of this kind. 
need occur. 

§ 19, Examples subject to the ambiguity now 



1 



I^CT. 19.] LOGIC OF CHANCS. 189 

under consideration, will doubtless seem perplexing to 

the student unacquainted with the subject They are, 

I think, quite irreconcileable with any other view of 

the science than that insisted on throughout this essay, 

Viz. that we are only concerned with averages. It 

will perhaps be urged, there are two different values 

of the man's life in these cases, and they cannot both 

be true. Why not ? The * value' of his life is simply 

the number of years to which men in his circumstances 

do, on the average, attain ; we have the man set before 

us under two different circumstances; what wonder, 

therefore, that these should offer different averages? 

In such an objection it is forgotten that we have had 

to substitute for the unattainable result about the 

individual, the really attainable result about a set of 

men as much like him as possible. The difficulty and 

apparent contradiction only arise when people will 

try to find some justification for their belief in the 

individual case. What can we possibly conclude, it 

may be asked, about this paiticular man John Smith ? 

Nothing whatever, I reply ; nor could we in reality 

draw a conclusion, be it remembered, in the former 

case, when we were practically confined to one set of 

statistics. There also we had what we called the 

'value' of his life, and since we only knew of one 

such value, we came to regard it as in some sense 

appropriate to him as an individual. Here we have 

two values, belonging to different series, and as these 



190 LOGIC OP CHANCE. [CHAP. VH. 

Talues are reallj different tbey seem discordant, but 
this complaint can only be made when we do what 
we have no right to do, viz. assign a value to the indi- 
vidual which shall admit of individual justification. 

§ 20. Is it then perfectly arbitrary what series we 
select by which to judge 1 By no means; I have 
been trying to show throughout that in choosing a 
series we must seek for one the members of which 
shall resemble our individual in as many of his attri- 
butes as possible, subject only to the restriction that 
it must be a sufficiently extensive series. What I 
mean is, that in the above case, where we have two 
series, we cannot fairly call them contradictory; the 
only valid charge is one of incompleteness or insuffi- 
x^iency, a charge which applies in exactly the some 
sense, be it remembered, to all statistics which com- 
prise genera unnecessarily wider than the species with 
which we are concerned. The only difference between 
the two different classes of cases is, that in the one 
instance we are on a path which we know will lead at 
the last, through many eiTors, to the truth (in the 
sense in which truth can be attained here), and we 
took it for want of a better. In the other ioBtuifie 
we have two such paths, perfectly different paths, but 
jBither of which will lead us to the truth as before. 
Contradiction can only seem to arise when it is at- 
tempted to justify each separate step on our paths^ 
as well as their ultimate conclusion. 



I 



SECT, 21.] LOGIC OF CHANGE. 191 

We may now see why the Eule of Succession i^ 
ambiguous as well as erroneous, as described in the 
last chapter. When we observe a succession of indi- 
vidual things or events of any kind, the number of 
terms in the succession will depend upon the number 
of properties we take into account. And as it seems 
4}iiite arbitrary how many of these properties we 
should take into account, the possible inferences we 
can draw will be various. 

§ 2L The foregoing results may be thus summed 
up : — 

Since the generalization of our statistics is found 
-to belong to Induction, this process of generalization 
may be regarded as prior to, or at least independent 
.^Probability. Wehvve, moreover, already discussed 
^ dlH^ter ni.) the step corresponding to what are 
termed immediste inferences, and (in Chapter rv.) 
that correspondiBg to syllogistic inferences. Our 
present position therefore is that in which we may 
consider ourselves in possession of any number of 
generalizations, but wish to employ them so as to 
make inferences about a given individual ; just as in 
one department of common logic we are engaged in 
finding middle terms to prove our argument. In this 
latter case the process is found to be extremely sim- 
ple, no accumulation of different middle terms can 
lead to any real ambiguity or contradiction. In Pro- 
bability, however, the case is different. Here, if we 



192 LOGIC OF CHANCE. [cHAP. YIi; 

attempt to draw inferences about the individual case 
before us, as often is attempted, — in the Eule of Sug<- 
oession for example, — we shall encounter the full foroe 
of this ambiguity and contradiction. Treat the quea- 
tion, however, fairly, and all difficulty disappears. 
Our inference really is not about the individuals as 
individuals, but about series or successions of theim. 
We want to know whether John Smith will die within 
the year ; this however cannot be known. But John 
Smith, by the possession of many attributes^ belongs 
to many different series. The multiplicity of middle 
terms therefore is what ought to be expected. We can 
know whether a succession of men, paupers, consump- 
tives, <ko. die within a year. We may make our se- 
lection therefore amongst these, and in the long ran 
the belief and consequent conduct of ourselves, and 
other persons (as described in Chapter iii.), will be 
capable of justification. With regard to choosing one 
of these series rather than another, we have two op^ 
posing principles of guidance. On the one hand, the 
^ore special the series the better ; for, though not 
more right, in the end, we shall thus be more nearly 
right all along. But, on the other hand, if we try to 
make the series too special, we shall generally meet the 
practical objection arising from insufficient statistica. 

§ 22. Throughout the discussions in this chapter 
it has been assumed that the common property which 
was observed, in different individualsi and was th<^ 



1 



SECT. 23.] THE LOGIC OF CHANCE. 193 

by the Inductive act generalized throughout a definite 

or indefinite class, is one about the existence and 

amount of which there could never be any doubt or 

dispute. Most of the examples commonly given, and 

the discussions to which they often lead, tend very 

much to confirm such an opinion. I have deferred 

all examination of this point up to the present mo* 

ment, in order not to break the line of enquiry; but 

it will be advisable now to devote some pages to 

ascertain how far the assumption spoken of above can 

be justified. The enquiry is, indeed, only indirectly 

connected with Probability, at least on the view of 

that science entertained in this Essay, but at the 

same time the connection though indirect is very 

close. Induction is involved in almost every example 

in Probability, and the nature of the relation between 

these sciences has been a source of so much error 

and confusion, that I have found it quite impossible 

to state accurately my own opinion about the latter 

science without making constant inroads into the 

former. It is the more necessary to state my opinion 

fully, in consequence of the great authority of Mr 

Mill being apparently in support of what I cannot 

but regard as an imperfect view of the subject. 

§ 23. It will be necessaiy to commence with a 
definition of Induction. Let us start with that of 
Mr Mill, which, though given by him in various 
forms, will be expressed, I think^ with substantial 

13 



194 THE LOGIC OF CHANCE. [CHAP. VII.; 

accuracy as follows : — " Induction is that net of the 
mind by which, from a cei-tain definite number of 
things or observations, we make an inference extend- 
ing to an indefinite number of them.'' In this defi- 
nition, which is in accordance with the investigation 
in the earlier part of this chapter, it is assumed that 
the data, viz. the limited number of things from which 
the formula starts, and on which it is grounded, are 
already clearly recognized. In every example, indeed, 
this recognition is almost necessarily presupposed be- 
fore the example can be stated. Whether we take 
the simple symbolic one, this A and that A, and 
so on, are X; therefore every ^ is X; or any special 
concrete instance that the Inductive Sciences mey 
offer, all practical difficulty which may have existed 
as to discovering and recognizing our A is omitted 
from view. But though this omission is possible in 
examples, it is scarcely possible in making original 
inferences. The objects from which our inference 
started as its basis must have been selected ; and 
since this selection was neither made at random nor 
perfonned for us by others, there must have been 
some principle of selection in our minds. The selec- 
tion may appear little more than a guess sometimes, 
but even then it is the sagacious guess of one whose 
mind has been disciplined to the work. 

§ 24. As most readers of these pages will know, 
Dr Whewell has strongly insisted on this selective or. 



1 



SECT. 24.] THE LOGIC OP CHANCE. 195 

as it may almost be termed, creative part of the pro* 
cess of Induction, and has applied a particular tech- 
nical expression to denote it* Being probably one of 
the few who have deduced their philosophical schemes 
from a laborious investigation of the processes by 
which science has been actually constructed, he has 
not unnaturally attached extreme importance to this 
paii) of the act of Induction, and has incorporated 
it into his definition of Induction. He sometimes 
uses rather strong expressions to describe it> but I 
think that what I have just mentioned is all that 
is meant when he speaks of the element which is 
introduced by the mind, and is not found in the 
things. The data for the Induction, therefore, have 
to be selected by a process which the common ex- 
amples tend to let drop out of sight. 

Let us take one of these examples for closer ex- 
amination, for instance, the familiar one: — This man 
is mortal, that man is mortal, and so on; therefore 
all men are mortal. Now here it is obvious that the 
* conception,' to use Dr Whe well's expression, is one 
with which we are already familiar. The collection 
of attributes which make up what we understand by 
humanity has been so constantly associated together, 
and this association has been so universally aided by 
common language, that nobody can see one of the 
objects which contain these attributes without having 
the class recalled to him, or at least without having 

13—2 



196 THE LOGIC OF CHANCE. [CHAP. YII. 

some of tbe objects which compose the class clearly 
separated off and brought before his mind. And 
similarly with regard to the attribute of mortality. 
The words 'man' and * mortal* suggest to every 
mind the appropriate conceptions. Now though, as 
we have already seen, and aa will be noticed again 
soon, there is still room even here for much ambiguity, 
this familiarity with the conceptions enormously di- 
minishes the difficulty of making inferences. It com- 
pletely alters the character of that process in fact; so 
that instead of being like a drive to the right point 
over the open plain, it now resembles the choice of 
one out of a limited number of ruts. 

§ 25. A slight modification of the example will 
make it wear a very different aspect. Let us try to 
find a case in which the conception is not marked out 
already for us by a word, and we shall see what a 
different relative importance is then assumed by the 
two parts of the Inductive process. There is some 
difficulty, it must be observed, in finding an example 
of the kind required, for unless there be a word 
already applied to the objects which we group toge- 
ther the example will become very awkward to state. 
But we may obviate the difficulty as follows. Let 
us take an example in which we, having the word, 
are already familiar with the conception, but suppose 
the inference to be made by people who have not 
the word, the person who makes the inference will 



} 



9SCT. 25.] THE LOGIC OF CHANCE. 197 

then labour under the difficulty mentioned, whilst we 
who stand by and criticise him will escape it. 

Let us take, then, the following example. This 
consumptive man, and that consumptive man, and 
so on, are short-lived; therefore, generally, all con- 
sumptive persons are short-lived. To us this example 
is as simple as the last, but suppose the inference 
to be made for the first time by some one amongst 
a people where consumption had not yet been suffi- 
ciently known to be marked out and named. The 
j>rocess will then, to them, assume a very different 
form. The cough, the shortness of breathing, &c., 
which the word at once suggests to us, will have 
to be singled out and distinguished ii'om a multitude 
of somewhat similar phenomena. There is room here 
for an almost infinite amount of observation and ex- 
periment. But now suppose that any one had got 
to this point, and, what is more, had connected the 
qualities of consumption and short-life in two or three 
instances. By this time he has gone partly through 
the process of Induction according to Dr Whewell, 
but he has only reached its threshold according to 
Mr Mill. But what more is there left for him to 
infer 1 Surely every one who had got to this point 
-would go on to infer, therefore all consumptive per- 
sons die soon. Whether they would be right or 
-wrong in so doing is of course immaterial to our pre- 
sent purpose. The in&rence would be made, and 



198 THE LOGIC OF CHANCE, [CHAP. VH. 

therefore what Mr Mill considers to be Induction 
would take place, so simply and certainly as to be 
performed almost unconsciously. 

§ 26. But there is still another source of doubt 
and confusion. It needs but a slight observation to 
perceive that even in the former example, that of 
man's mortality, the inference is not so simple as it 
is made to appear. With all the immense help of 
the previous inductions and observations of others, 
which are fixed and perpetuated for us in language, 
there is still a degree of arbitrariness in the process as 
it is given in common examples. If we try to place 
ourselves in the position of one making the inference 
for the first time we shall see that we might then be 
involved in serious perplexity. It should be borne 
in mind that when we state the grounds of the in- 
ference in the form, this man is mortal, and that man 
is ; we are presupposing that the observer has already 
not merely distinguished the class 'man,' but distin^ 
guished it as appropriate to his immediate purpose. 
Whereas the grounds of his inference were in reality 
certain objects. It is true that these objects belong to 
the class man, but, as I have pointed out, they belong 
also to an indefinite number of other classes as well ; 
to classes within that of man; for example, Greek, 
European, &c, : to classes without that of man ; for 
example, mammalia, animal, organized being. As it 
happens, the observer would have been right in hia 



J5BCT. 27.] THE LOGIC OP CHANCE. 199 

inference wbicliever of these classes and corresponding 
conceptions he might have selected ; but such con- 
siderations show us that there was scope for great 
ingenuity and for considerable effort of mind in a 
preliminary process, which, according to Mr Mill, is 
no part of Induction. 

When therefoire Mr Mill states, as he does in 
another part of his volume, that the mortality of 
John, Thomas, &c, is the whole evidence for the death 
of Socrates, and for that of men in general, he is, as it 
appears to me, very much underrating the complexity 
of the problem. These objects belonged, no doubt, 
to the class mauy but they belonged to an indefinite 
number of other classes also. What made us draw 
the line just where we did 1 Why did we not go 
farther? Js not our belief strengthened by the death 
of animals? Is it not influenced by that of organ- 
ized beings generally 1 The moment we admit the 
question to be so indefinite as this, we see that so 
far from the arbitrary selection of instances, which 
was given to us, being the whole evidence, it is but a 
fragment of the evidence. Analogies of every con- 
ceivable amount of strength press in upon us from 
every side, and help to swell or diminish the degree 
of our belief in the final result. 

§ 27. The bearing of all this upon rules of dis- 
covery is obvious. In the precise form in which 
such rules are commonly stated in Probability it is 



200 THE LOGIC OF CHANCE. [CHAP. VII. 

supposed that when an event has been observed to 
happen in a certain way, or an object to possess a 
certain attribute, a given number of times in' succes- 
sion, we are able to assign the degree of belief which 
should be entertained as to the recurrence of this 
event or property. The rule, in this particular form, 
was fully examined and shown to be Mse, but we 
are now. able to perceive that independently of its 
falsity any rule of the kind is impracticable. It is 
not merely that we are forbidden to say that, be- 
cause three A's have been found to be X it is there- 
fore 4 to 1 that the next A will be X. Before the 
problem could be stated even, a process that is ofben 
veiy difficult and tedious has to be gone through. 
A and X have to be recognized and distinguished, 
and if the conception is one with which we are not 
familiar this distinction will be very partial and 
progressive. And when the conception is formed, 
we should generally find it impossible to limit, our 
grounds of belief to the objects denoted by this con- 
ception; an indefinite number of other conceptions 
would all seem to have an almost equal claim to 
acceptance. It would be quite arbitrary to reject 
entirely all these other conceptions, and if we do not 
reject them the data on which our belief is founded 
become almost infinite in number, and therefore vague 
in value. Things and qualities, it must be remem- 
bered, are far from being so sharply discriminated 



1 



I 



SECT. 27.] THE LOOIO OF CHANCE. 201 

from one another as are letters and symbols. We 
have observed consumptiveness in Englishmen ; is it 
quite the same quality in people of other countries 1 
Is there not something resembling it in some animals ] 
We may give the quality the same name, but in doing 
so we must remember that it possesses eyeiy con- 
ceivable amount of variation in degree ; and if we 
are reasoning with any accuracy this variation should 
influence the amount of our assent. Moreover, in 
addition to the indefiniteness caused by the object 
being thus included in a number of classes which 
point with varying degrees of force towards the same 
inferences, there is (as already pointed out) the em- 
barrassment caused by its being included in classes 
which point towards conflicting inferences. I do not 
see then how such rules of Anticipation can ever be 
more than a collection of hints and suggestions for 
making judicious guesses. When any one has ob- 
tained the conception of consumption, and has ob- 
served that it is accompanied in certain known cases 
by short-life, he may and probably will go on to the 
inference that it will be so in more cases, if not in 
aJl. The proposition, therefore, All consumptive per- 
sons die early, is gradually borne in upon him. He 
cannot point to any limited, definite number of in- 
stances on which this Induction rests ; it is supported 
rather by an indefinite number of analogies more or 
less close or remote. But the first point at which 



202 THE LOGIC OP CHANCE. [cHAP. VH. 

anything like scienti£c reaaoniiig can commence is 
the point at which this proposition is obtained. 

Precisely similar is the difficulty, and equally grsr- 
dual the process, of obtaining any one of the pro- 
portional propositions with which we have been con- 
cerned. But when such propositions are obtained we 
are then in a position to draw certain inferences by 
strict rules. If they are universal propositions they 
belong to ordinaiy Logio, if proportional to Proba- 
bility. 



CHAPTER VIII. 
ON DIEECT AND INVERSE PROBABILITY. 

§ 1. We must now take some notice of a distinc* 
tion, commonly drawn in works on this subject, be- 
tween what is called Direct and Inverse Probability. 
The distinction is thus introduced by De Morgan* : 
" In the preceding chapter we have calculated the 
chances of an event, knowing the circumstances under 
which it is to happen or fail. We are now to place 
ourselves in an inverted position : we know the event, 
and ask what is the probability which results from the 
event in favour of any set of circumstances under 
which the same might have happened." 

On the principles of the science involved in the 
definition which was discussed and adopted in the 
earlier chapters of this work, the reader will easily 
infer that no such distinction as this can be regarded 
as fundamental. One common feature was traced in 
all the subjects which were to be referred to Pro- 
bability, and from this feature the possible rules of 
inference were immediately derived. All other dis- 
tinctions are merely of arrangement or management. 

* £»tay on Probabilities, p. 53. 



204 THE LOGIC OP CHANCE. [CHAP. VIH. 

The apparent importance of the distinction under 
discussion arises entirely, I cannot but think, from 
the attempt to force the Calculus of Probability upon 
a class of subjects which do not properly belong to 
it, and about which an arbitrary supposition must be 
made before the rules admit of application. 

§ 2. This will be best seen by the examination 
of special examples; as any, however simple, will 
serve our purpose, let us take the two following :— 

(1) A ball is drawn from a bag containing nine 
black balls and one white j what is the chance of its 
being the white ball] 

(2) A ball is drawn from a bag containing ten 
balls, and is found to be white; what is the chance 
of there having been but that one white ball in the 
bag? 

The class of which the first example is a simple 
instance has been already abundantly discussed. The 
interpretation of it is as follows: If balk be con- 
tinually drawn and replaced, the proportion of white 
ones to the whole number drawn will tend towards 
the fraction ^. The contemplated action is a single 
one, but we view it as one of the above series ; at 
least our o^nnion is formed upon that assumption. 
We conclude that we are going to take one of a series 
of events which may appear individually fortuitous, 
but in which in the long run those of a given kind 
are one-tenth of the whole; this kind (white) is then 



1 



BECT. 3.] THE LOGIC OF CHANCE* 206 

siDgled out by anticipation. By stating that its 
chance is i^, we merely mean to assert this physical 
fact, together with such other mental facts, impresh 
sions, inferences, &c., as may be properly associated 
with it. 

§ 3. Have we to interpret the second example 
in a different way? Here also we have a single 
instance, but the nature of the question would seem 
to decide that the only series to which it can pro- 
perly be referred is the following : — Balls are continu- 
ally drawn from different bags each containing ten^ 
and are always found to be white ; what is ultimately 
the proportion of cases in which they will be found 
to have been taken from bags with only one white 
ball in them? Now it was shown in the last 
chapter that time has nothing to do with the 
question ; omitting therefore the consideration of this 
element, we have for the two series from which our 
opinions in these two examples respectively are to 
be formed : — (1) balls of different colours presented 
to us in a given ratio ; (2) bags with different con- 
tents similarly presented. From these data we have 
to assign their respective weight to our anticipations 
of (1) a white ball; (2) a bag containing but one 
white balL So stated the problems would appear to 
be formally identical, the only difference being in the 
matter. Theory therefore would recognize no dis- 
tinction between them. 



206 THE LOGIC OF CHANCE. [CHAP. Vim 

When, however, we begin the practical work of 
solving them we perceive a most important distinc- 
tion. In the first example there is not much that 
is arbitrary; balls wonld under such circumstance 
really come out in somewhat the proportion expected. 
Moreover, it does not seem an unfair demand to 
say that the balls are to be * well-mixed ' or * fairly 
distributed,' or to introduce any of the other condi- 
tions by which, under the semblance of judging d 
prioriy we take care to secure our prospect of a series 
of the desired kind. But we cannot say the same in 
the case of the second example. 

§ 4. The line of proof by which it is generally 
attempted to solve the second example is of this kind ; 
— It is shewn that there being one white ball for 
certain in the bag, the only possible antecedents are 
of ten kinds, viz. bags, each of which contains ten 
balls, but in which the white balls i^nge respectively 
from one to ten in number. This of course limits 
the kind of terms to be found in our series. But 
we want more than this limitation; we must know 
the proportions in which these terms are ultimately 
found to arrange themselves in the series. Now this 
requires an experience about bags which is not given 
to us. If therefore we are to solve the question at 
all we must make an assumption ; let us make the 
following; — tliat each of the hags described above occurs 
eqiuiUy o/ten, — ^and see what follows. The bags being 



SECT. 5.] THE LOGIC OF CHANCE. 207 

drawn from equally often, it does not follow that 
they will yield equal numbers of white balls. On 
the contrary they will, as in the last example, yield 
them in direct proportion to the number of such 
balls which they contain. The bag with one white 
and nine black will yield a white ball once in ten 
times; that with two white, twice; and so on. The 
result of this, it will be easily seen, is that in 100 draw- 
ings there will be obtained on the average 55 white 
balls. Now with those dm wings that do not yield 
white balls we have, by the question, nothing to do, 
for that question, as it was originally stated, postu- 
lated* the drawing of a white ball. The series we 
want is therefore composed of those which do yield 
white. Now what is the additional attribute which 
is found in some member of this series, and which 
we mentally anticipate? Clearly it is the attribute 
of having been drawn from a bag which only con- 
tained one of these white balls. Of these there is 
out of the 55 drawings but one. Accordingly the 
required chance is ■^. That is to say, that the white 
ball will have been drawn from the bag containing 
only that one white, once in 55 times. 

§ 5. Now, with the exception of the passage in 
italics, the process here is precisely the same as in 
the other example; it is somewhat longer only be- 
cause we are not able to appeal immediately to ex- 
perience, but are forced to try and|]deduce^ what the 



208 THE LOGIO OF CHA17CE. [CHAP. TIIL 

result will be, though the validity of this deduction 
itself rests, of course, upon experience. But the 
above passage is a very important one. It is scarcely 
necessary to point out how entirely arbitrary it is. 
The nature of the assumption is commonly disguised 
by saying that, where we have no reason to expect 
one kind of bag rather than another it is only reason- 
able to regard all possible kinds as equally likely. 
But, as I have before insisted, this phrase 'equally 
likely ' is one that must submit itself to explanation. 
If any one replies that he means nothing more by 
it than that he expects the events equally, he is at 
liberty to do so; but he should remember that he 
is applying the words entirely to his state of mind, 
and avowing that they have no necessary connection 
with experience. If they are to have such a con- 
nection he can only mean that the events do happen 
equally often ; for our opinion about the events being 
equally likely is simply worthless, indeed should 
rather be called a guess, unle^ we have good reason 
to believe that it is really in harmony with experi- 
ence. Now would it not somewhat surprise an un- 
prejudiced person if his assent were demanded to the 
proposition that, in his experience in this world, 
bags of the kind described above occurred with equal 
frequency f or even if he had merely to assert that 
it was his honest conviction that they would do so 1 
We may assume, as in the common hypothesis, that 



1 



8BCT. 6.] LOGIC OF CHANCE. 209 

such are the conditions under which the course of 
nature is carried on; but it is only fair that the 
assumption should be openly recognized. 

§ 6. I will now take an instance which shall be 
entirely from nature, so as not to require an arbi- 
trary supposition of the kind just discussed; it will 
then be seen that this distinction between Direct and 
Inverse Probability disappears altogether, or resolves 
itself into one of time merely, which, as was shown 
in the last chapter, is entirely alien to our subject. 

(1) What is the chance that if A. B. dies he will 
die of typhus fever 1 

(2) A. B. is dead ; what is the chance that h^ 
died of typhus ? 

If we refer to the statement of De Morgan at 
the commencement of this chapter, we shall see that 
these two examples undoubtedly fall under the head 
of what he there defines as Direct and Inverse Pro- 
bability, If therefore the distinction breaks down 
in this case it cannot be an essential distinction, for 
this cannot be affected by the simplicity of the ex- 
ample chosen. Kow a moment's consideration will 
show that, considered as questions in Probability, 
these two examples are absolutely identical In each 
alike the enquiry is. What weight should be attached, 
prior to examination of the case^ to the anticipation 
that a man's death is caused by typhus fever ? The 
statistical data by which this question is to be an- 

U 



210 LOGIC OF CHANCE. [CHAP. VUI. 

swered, of course, are also the same, viz. Of the total 
number of deaths, what is the proportion caused by 
that disease? There is to be sure this distinction, 
which may be important in other respects, that in 
the first example what is an anticipation to us is also 
an anticipation to all other people, owing to the &ct 
of the man being yet alive ; whereas in the second 
example, the man being now dead, other people know 
already, and we might probably if we took the trou- 
ble, ascertain, what the cause of death really was. 
But when these questions are treated as examples 
in Probability, it would be wandering from the point 
to found a distinction upon the difference that would 
exist between them if they were not so treated. 

§ 7. Considered, therefore, as a portion of the 
theory of the subject, the distinction between Direct 
and Inverse Probability must be abandoned; but 
the discussion of it may serve to direct renewed at- 
tention to another and far more important distinction. 
It will remind us that there is one class of examples 
to which the calculus of Probability is rightfully 
applied, because statistical data are all we have to 
judge by ; whereas there are other examples in regard 
to which, if we will insist upon making use of these 
rules, we may be deliberately abandoning the oppor- 
tunity of getting far more trustworthy information 
by other means. This is a point to which reference 
will be made again in a future chapter. 



SECT. 7.] LOGIC OP CHANCE. 211 

It will show too, with respect to some examples 
of the latter kind, how much there is that is wholly 
arbitrary in the ordinary treatment of the subject. 
"Writers will apply their rules to instances in which 
not merely other evidence .is at hand, but in which 
statistics are not There being positively no ex- 
perience of any kind upon the subject, either we have 
to divorce the science from expeiience and make it 
therefore the study of our own unauthorized im- 
pressions, or else we find ourselves at a loss. But as 
it is a hard thing to be deprived of the right of 
forming a positive opinion merely because we happen 
to have no grounds on which to rest it, we are driven 
to assume for ourselves a fictitious experience, as 
already described, which opens the way at once to 
confusion and dispute. 



U—2 



CHAPTER IX. 

CRITICISM OF SOME COMMON CONCLUSIONS IN 
PROBABILITY. 

§ 1. The view taken in the seventh chapter of 
the connection between Probability and Induction 
will suggest considerable doubts as to the validity 
of some commonly accepted conclusions of the former 
science. There is, for instance, a difficulty frequently 
anticipated and discussed by writers upon the subjeetb 
They assure those who are beginning their studies 
in it that no anterior improbability is a bar to a thing 
happening in time ; in fact, if there be only an impro- 
bability, and not an impossibility, this insures that 
the thing shall happen. In other words, whatever 
be the odds against an event, if we only go on long 
enough the event is sure to come to pass at length. 
That three pence should all give heads is not so 
likely as that one should do so ; that four should give 
heads is still more unlikely, and so on; but all these 
events occur in their due time. Proceeding in this way 
it is inferred that ten heads from as many pence, though, 
of course, very unlikely, will yet be found in its due 
proportion of instances ; viz. once in 1024 times. It 



SECT. 2.] LOOIC OF CHANCE. 213 

is assumed that snch an inferexice will be receired 
with some doubt, and arguments are given to convince 
the hesitating student of its truth. He is reminded 
that improbability, however extreme, is not only dif^ 
ferent from impossibility, but positively excludes it. 

As an explanation of part of the difficulty, the 
above remarks are very useful and to the point. 
There are many persons to whom a good deal of illus- 
tration will be necessary before they can realize that 
to say that the chances are enormously against a thing 
is only another way of saying that the thing will 
really happen occasionally though very rarely. I 
confess however that this answer appears to me to 
leave the main difficulty untouched, by assuming the 
real questions at issue. If the chances be 1023 to 1 
against the above-mentioned occurrence, it imdoubt- 
edly will happen once in 1024 times. This is the 
meaning of the statement ; it is in fact but another 
form of expression. But are the chances 1023 to 1 
against the event? This is the point upon which 
we are now going to give a brief discussion. 

§ 2. To the question, when stated in this form, 
the very summaiy answer may perhaps be given, that 
to doubt the result is to doubt the truth of mathe- 
matics, with the proviso, of course, that we are 
talking of an ideal penny. .It has been stated already 
that a penny may be idealized, but that to talk of 
performing this process upon 'randomness' and the 



214 LOGIC OP CHANCE. [CHAP. IX^ 

other conditions which are all equally coefficients of 
the cause by which the effect is produced, is to use 
words with little meaning; at least the meaning 
can only be that the effect, t. e, in this case the suc- 
cession of throws, is apportioned in the way sup- 
posed to be assigned by the chance, so that we are 
in a circuitous way talking about the succession it- 
self It is of no use therefore to speak of ideal 
probabilities ; we must adopt one or other of these 
alternatives; — either we are working out a purely 
aiithmetical sum of combinations and permutations, 
choosing to apply to the results the names of 'chances ;' 
or we are making inferences about the actual be- 
haviour of objective things, in which case though 
our results may fail of perfect accuracy, they must 
at least have a fair foundation of fact. If the former 
alternative be adopted, I have nothing more to say, 
as the present is not a treatise upon any branch of 
mathematics. But if the latter be adopted, the ques- 
tion may be fairly asked, Does the formula give ao* 
curate results ? 

§ 3. We will give the rule the considerable ad- 
vantage of assuming that the pence are perfect, so 
that in the long run they show no preference for 
either head or tail ; the question then remains. Will 
the repetitions of the same, face obtain the proportional 
shares to which they are entitled, if the theory be 
correct 1 We intend to refer to high numbers, bat 



SECT. 4.] LOGIC OP CHANCE. 2 15 

the illustration will be simpler if we begin with a 
small one, for example, a repetition of two. Putting 
then, as before, for the sake of brevity, H for head, 
and HH for heads twice running, we are brought 
to this issue ; — Given that the chance of H is J, 
does it follow necessarily that the,chance of HH (with 
two pence) is ^ ? To say nothing of * H ten times ' 
occurring once in 1024 times (with ten pence), need 
it occur at all? The mathematicians, for the most 
part, seem to think that this conclusion follows neces- 
sarily from first principles; to me it seems to rest 
upon no more certain evidence than a reasonable ex- 
tension by Induction. 

§ 4. Taking then the possible results which can 
be obtained from a pair of pence, what do we find? 
Four different results may follow, namely, (1) HT. 
(2) HH. (3) TH. (4) TT. If it can be proved 
that these four are equally probable, that is, occur 
equally often, the commonly accepted conclusions will 
follow, for a precisely similar argument would apply 
to all the larger numbers. 

The proof usually advanced makes use of what 
is called the Principle of Sufficient Reason. It takes 
this form ; — Here are four kinds of throws which 
may happen ; once admit that the sepai*ate elements 
of them, namely, H and T, happen equally often, and 
it will follow that the above combinations will also 
happen equally often, for no reason can be given in. 



216 LOGIC OF CHANCE* [cHAP. UC 

&voar of one of tbem that would not eqiudly hold in 
favour of the others. 

To this mode of argamoit reference has already 
been made more than once. Surely some stronger 
reasons ought to be given for believing in a result than 
an inability to see why it should not come as soon 
as anything else. But, even if the rule were ad- 
mitted to be valid, I think it is &r too readily as- 
sumed that the rule would be applicable to prove 
the result in question. Let us examine the proof 
somewhat more closely. There are four different re- 
sults which may happen. In the symbolic repre- 
sentation of them given above they appear to differ 
from one another, for the letter H is different from 
the letter T ; but this difference is of course entirely 
owing to our notation, the sides themselves of the 
pence having been expresidy idealized into absolute 
similarity*. As between the single faces therefore 
H and T, the rule if sound would be applicable ; no 

* I am endeavouring to treat this rule of Suffident Reason 
in a way that shall be legitimate in the opinion of those who 
accept it, but there seem very great doubts whether a contradic- 
tion is not involved when we attempt to extract results from it. 
If the sides are absolutely alike, how can there be any difference 
between the terms of the series? The succession seems then 
reduced to a dull uniformity, a mere iteration of the same thing 
many times ; the series we contemplated has disappeared. If the 
sides are not absolutely alike, what becomes of the applioability 
of the rule? 



P.S 



SECT. 4.] LOGIC OP CHANCK 217 

y^ distinction can be observed between these except the 
merely apparent one arising out of our own nota- 
^ tion. To suppose H therefore to occur more often 
^ than T, namely head to occur more often than tail, 
Ijj would be an infraction of the rule. But it seems 
yf^ to be too hastily assumed that the same must be 
jij. the case as between the pairs of faces already men* 
j^ tioned. 

,^ To a certain extent I admit the validity of 

■ji the rule for the purpose. In the series given above 
^ it would be valid to prove the equal frequency of 
f (1) and (3) and also of (2) and (4) ; for there is no 

's difference existing between these pairs except what 

3 is introduced by our own notation. TH is the same 

T as HT, except in the order of the occurrence of 

i the symbols H and T, which we do not take into 

i account. But either of the pair (1) and (3) is dif- 

ferent from either of the pair (2) and (4). Trans- 
pose the notation and there would still remain here 
a distinction which the mind can recognize. A suc- 
cession of the same thing twice running is distin- 
guished from the conjunction of two different things 
by a distinction which does not depend upon our 
arbitrary notation only, and would remain entirely 
unaltered by a change in this notation. The principle 
therefore of Sufficient Beason, if admitted, would only 
prove that doublets of the two kinds, for example, 
(2) and (4), occur equally often, but it would not 



218 LOGIC OF CHAXCS. [cHAP. IX. 

prove that they must each occur once in four times. 
It cannot be proved indeed in this way that they 
need ever occur at all. 

§ 5. The formula, then, not being demonstrably 
h priori, (as might have been concluded,) can it be 
obtained by experience ) To a certain extent it can ; 
the present experience of mankind in pence and dice 
seems to show that the smaller successions of throws 
do really occur in about the proportions assigned by 
the theory. But how nearly they do so, no one can 
say, for the amount of time and trouble to be ex- 
pended before we could feel that we have verified 
the fact, even for small numbers, is very great, whilst 
for large numbers it would be simply intolerable. 
The experiment of throwing often enough to obtain 
'heads ten times' has been actually performed by 
two or three persons, and the results are given by 
De Morgan. This, however, being only sufficient on 
the average to give 'heads ten times' a single chance, 
the evidence is very slight ; it would take a consider- 
able number of such experiments to set the matter 
at rest. 

Any such rule, then, as that which we have just 
been discussing, which professes to describe what will 
take place in a long succession of throws, is only con- 
clusively proved by experience within very narrow 
limits, that is, for small repetitions of the same face ; 
within limits less narrow, indeed, we feel assured 



1 



SECT. 6.] LOGIC OF CHANCE. 219 

that the rule cannot be flagrantly in error, otherwise 
the variation would be almost sure to be detected. 
From this we feel strongly inclined to infer that the 
same law will hold throughout In other words, we 
are inclined to extend the rule by Induction and 
Analogy. Still there are so many instances in nature 
of laws which hold within narroy limits but get 
egregiously astray when we attempt to push them to 
gi'eat lengths, that we must give at best but a qualified 
assent to the truth of the formula. I breathe no 
Suspicion, let it be observed, against the integrity of 
the mathematics introduced, but only deny that they 
can be taken as authoritative about the physical facts 
to which they are applied. In other words, we cannot 
' be sure that we have obtained the right mathematical 
formula. 

§ 6. The object of the above reasoning is simply 
to show that we cannot be certain that the rule is 
true. Let us now turn for a minute to consider the 
causes by which the succession of heads and tails is 
produced, and we may perhaps see reasons to make 
us still more doubtful. 

It was shown in Chapter ii. that in calculating 
probabilities d priori, as it is called, we were only 
able to do so by introducing restrictions and sup- 
positions which were in reality equivalent to assuming 
the expected results. We used words which in strict- 
ness mean. Let a given process be performed; but 



220 LOGIC or OHANCE. [chap. IX. 

an analysis of our language, and an examination of 
various tacit suppositions whicli made themselves felt 
the moment they were not complied with, soon showed 
that our real meaning was. Let a series of a given kind 
be obtained ; it is to this series only, and not to the 
conditions of its production, that all our subsequent 
calculations properly apply. In the present instance 
this transformation from an actual and limited to ai^ 
ideal and unlimited series cannot be allowed, for 
our express object is to examine into the validity of 
these suppositions. The physical process being per- 
formed, we want to know whether anything resem- 
bling the contemplated series really will be obtained. 

Now if the penny were invariably set the same 
side uppermost, and thrown with the same velocity 
of rotation and to the same height, &c. — ^in a word, 
subjected to the same conditions, — it woidd always 
come down with the same side uppermost. Practi- 
cally, we know nothing of this kind occurs, for the 
individual variations in the results of the throws 
are endless. Still there will be an average of these 
conditions, about which the throws will be found, 
as it were, to cluster much more thickly than else- 
where. We should be inclined therefore to infer that 
if the same side were always set uppermost there 
would really be a disturbance in the series which 
we ordinarily look for. In a very large number of 
throws we should probably begin to find, under such 



SECT. 6.] LOGIC OP CHANCE. 221 

drcumstances, tliat either head or tail was having 
a preference shown to it. If so, would not similar 
effects be found to be connected with the way in 
which we started each successive 'pair of throws % 
Accoixiing as we chose to make a practice of putting 
HH or TT uppermost, might there not be a dis- 
turbance in the proportion of successions of two heads 
or two tails 1 Following out this train of reasoning 
it would seem to point with some likelihood to the 
conclusion that in order to obtain a series of the kind 
we expect, we should have to dispose the antece- 
dents in a similar series at the start. The changes 
and chances produced by the act of throwing might 
introduce infinite individual variations, and yet there 
might be found, in the very long run, to be a close 
similarity between these two series. 

This is, to a certain extent, only shifting the dif- 
ficulty, I admit; for the claim formerly advanced 
about the possibility of proving the proportions of 
ihe throws in the former series, will probably now 
be repeated in &vour of those in the latter. Still 
the question is very much narrowed, for we have 
reduced it to a series of volurUary acts; a man may 
put whatever side he pleases uppermost He may act 
consciously, as I have said, or he may think nothing 
whatever about the matter, that is, throw at random ; 
if so it will probably be asserted by many that he 
will involuntarily produce a series of the kind in 



223 LOGIC OP CHANCE. [CHAP. IX. 

question. It may be so, or it may not ; it does not 
seem that there are any easily accessible data by 
which to decide. All that I am concerned with here 
is to show the likelihood that the commonly received 
result does in reality depend upon the fulfilment of 
a certain condition at the outset, a condition which 
it is certainly optional with any one to fulfil or not 
as he pleases. The small numbers doubtless will take 
care of themselves, owing to the infinite complica- 
tions produced by the casual variations in throwing ; 
but the large ones may suffer, unless their interest 
be consciously or unconsciously regarded at the 
outset. 

An illustration may serve at once to explain and 
give support to the above view. Suppose that on a 
chess-board a number of distinct heaps of sand are 
arranged. Let the board be sharply struck from 
beneath ; the grains will fly up and then settle down 
again ; their arrangement being considerably disturbed 
by the process. But still there will be distinguish- 
able traces of their former distribution ; unless there 
were a good many large heaps before, we shall not 
find them afterwards. Single grains will be scattered 
all about, but the clusters will be less disturbed 
either in position or relative magnitude; and thus 
the former arrangement will really be found again 
on the whole, though disturbed and modified. Some- 
thing of this disturbance, and no more, may be pro- 



1 



SECT. 7.] LOGIC OF CHANCE. 223 

duced in the tossing up of a penny. In a very large 
number of throws we may find the order at the out- 
set reproduced with infinite individual variations ; 
and thus the long successions may not turn up in 
the end, at least not in their right proportion, un« 
less we oui*selves put them there in the beginning. 

§ 7. The advice ' only try long enough, and you 
will sooner or later get any result that is possible,' 
is plausible, but it rests only on Induction and Ana- 
logy ; mathematics do not prove it. As has been so 
often stated, there are two distinct views of the sub- 
ject. Either we may, on the one hand, take a series 
of symbols, call them heads and tails; HT; &c.; 
and make the assumption that each of these, and 
each pair of them, and so on, recurs in the long 
run with a regulated degree of frequency. (All these, 
it is to be observed, being perfectly distinct assump- 
tions.) We may then calculate their combinations 
and permutations, and the consequences that may 
be drawn from the data assumed. This is a purely 
algebraical process ; it is in£Edlible ; and there is no 
limit whatever to the extent to which it may be 
carried. 

This view may be, and undoubtedly should be, 
pothing more than the expression of what I have 
called the substituted or idealized series which gene- 
rally has to be introduced as the basis of our cal- 
culation. The danger to be guarded against is that 



224 LOGIC OP CHANCE. [CHAP. IX. 

of regarding it too purely as an algebraical concep- 
tion, and thence of sinking into the very probable 
errors both of too readily evolving it out of our own 
consciousnesss and too freely pushing it to unwarrant- 
ed lengths. 

Or on the other hand, we may consider that we are 
treating of the behaviour of things; — balls, dice, births, 
deaths, (&c. ; and drawing inferences about them. 
But, then, what were in the former instance allowable 
assumptions, become here propositions to be tested 
by experience. Now the whole theory of Probabi- 
lity as a practical science, in fact as anything more 
than an algebraical truth, depends of course upon 
there being a close correspondence between these two 
views of the subject, in other words, upon our sub- 
stituted series being kept in accordance with the 
actual series. Experience abundantly proves that, 
between considerable limits, in the example in ques* 
tion, there does exist such a corresjiondence. But 
let no one attempt to enforce our assent to every 
remote deduction that mathematicians can draw from 
their formulae. When this is attempted the distinc- 
tion just traced becomes prominent and important, 
and we have to choose our side. Either we go over 
to the mathematics, and so lose all right of discussion 
about the things; or else we take part with the 
things, and so defy the mathematics. We do not 
question the formal accuracy of the latter within 



1 



SECT. 8.] W)GIC OF CHANCE. 225 

their own province, but either we dismiss them as 
irrelevant, as applying to data of whose correctness 
we cannot be certain, or we take the liberty of re- 
modelling them so as to bring them into accordance 
with facts. ' 

§ 8. The extreme importance of obtaining a clear 
apprehension of the above distinction, has induced 
me to devote what might seem needless trouble to 
the illustration of it. A single example will serve 
to show the conclusions to which some thinkers have 
been led by consistently working out the data they 
have adopted. M. Quetelet, in his work on Proba- 
bilities, has didcussed* ^^the determination of the la/w 
of ocdtarence of two kinds of events, the dumces of whidh^ 
wre perfectly equals and which may happen either sepa- 
rately or simultaneously, but in different combinations. '*> 
The first half of this sentence is perfectly plain ; it 
means that the two kinds of events do, on the average, 
occur equally often. The latter part however is some- 
what obscure ; it appears to assume nothing, talking 
only about the way in which the events may happen. 
As would easily be seen, however, on examination, it 
introduces a very definite siipposition as to how they 
wUX happen; namely that, in accordance with the as- 
sumption criticized in this chapter, all the different 
combinations of the same number will occur equally 

* Qaetelet On Probabilities, by 0. G. Bownes, p. 6i. The 
italics are my own. 

15 



226 LOGIC OP CHANCE. [cHAP. IX, 

often. He selects the example of births and deaths 
as found succeeding one another in a register. He 
assumes very justly (the number of males and females 
being equal) that the chance of any one entry being 
male is one half. Then follows the next step, that the 
chance of having two males succeeding is one fourth. 
I have endeavoured to show that this is a distinct sup* 
position, which cannot certainly be deduced from the 
phrase in italics By following out the above process, 
the conclusion is arrived at that once in a certain de- 
terminate number of times we shall find the deaths of 
ten males happening successively. Thinking it pos- 
sible that one might like to know " how far experience 
justifies the calculation," and being a humourist as 
well as a mathematician, he remarks that the process 
of actually consulting the registers themselves would 
be 'Hedious," and that he will therefore resort to 
'' experiments more expeditious and quite as conclu- 
sive." He therefore puts forty black and forty white 
balls into a bag, proceeds to draw them, and to note 
the successions of each colour that come out^ and this 
is supposed to prove that men and women will die in 
certain proportions. If by men and women be meant 
black and white balls, I have no objections to offer ; 
but if the words denote anything more, one might be 
inclined to demur to some of his conclusions. I am 
quite aware that any hesitation to accept these conclu- 
sions would be met by the enquiry^ whether it is 



1 



SECT. 8.] LOGIC OF CHANCE. 227 

doubted that the events in question are independent, 
and their individual occurrence equally probable. 
This word ' independence ' has already been discussed ; 
under an appearance of specious modesty it really 
makes very extensive claims ; I can only say therefore 
that in the sense which must be assigned to the word, 
to justify the consequences which are commonly sup- 
posed to follow from it^ it could hardly be proved that 
the events are independent. Under ordinary circum- 
stances no perceptible deflection from the theory might 
be observed, but on the rare occasions on which any 
large number of one sex did happen to die in succes- 
sion it is quite possible that this might introduce a dis- 
turbance amongst the proportions of the deaths which 
succeeded such an occurrence. 



15—2 



1 



CHAPTEE X. 

THE APPLICATION OF PROBABILITY TO 
TESTIMONY. 

§ 1. On the priDciples which have been adopted and 
adhered to in this work, it will easily be seen that 
several classes of problems will have to be excluded 
from the science of Probability which may seem to 
have acquired a prescriptive right to admission. The 
most important, perhaps, of these refer to what is 
commonly called the credibility of testimony. Almost 
every treatise upon the science contains a discussion 
of the principles according to which credit is to be 
attached to combinations of the reports of witnesses of 
various degrees of trustworthiness, or the verdicts of 
juries. A great modem mathematician, Poisson, 
has written an elaborate treatise expressly npon this 
subject; whilst a considerable portion of the works of 
Laplace, De Morgan, Quetelet, and others, is devoted 
to an examination of similar enquiries. It would be 
presumptuous to differ from such authorities as these, 
except upon the strongest grounds; but I confess that 
the extraordinary ingenuity, research, and mathemati- 
cal ability which have been devoted to these problems, 
considered as questions in Probability, fail to convince 



SECT. 3.] LOGIC OF CHANCE. 329 

me that they ever ought to have been so considered. 
I proceed to give the grounds for this opinion, 

§ 2. It will be remembered that in the course of 
the chapter on Induction we entered into a detailed 
investigatioii of the process required when, instead of 
the appropriate series from which the deduction was 
to be made being set before us^ the individtuU pre- 
sented himself and the task was imposed upon us of 
selecting the requisite series. At such a stage w<0 may 
of course assume that the preliminary process of ob* 
taining the statistics which are extended into these 
aeries has been already performed; we may suppose 
therefore that we are already in possession of a quan- 
tity of series, our only doubt being as to which of 
them we should then employ. ^ This selection was 
shewn to be to a certain extent arbitrary; for, owing 
to the &ct of the individual possessing a large number 
of different properties, he became in consequence a 
member of different series, which might present differ-^ 
ent averages. We must now examine somewhat more 
fully than we did before the practical conditions under 
which any difficulty arising from this source ceases to 
be of importance. 

§ 3. One condition of this kind is very simple and 
obvious. lb is that the different statistics with which 
we are presented should not in reality offer materially 
different results. If, for instance, we were enquiring 
into the probability of a man aged forty dying witliin 



23Q LOGIC OF CHANCE. [cHAP. X. 

the year, we might if we pleased take into account the 
fact of his having red hair, or heing bom in a certain 
county or village. Each of these circumstances would 
serve to specialize the individual, and therefore to re- 
strict the limits of the statistics which were applicable 
to his case. But the consideration of such qualities 
as these would either leave the average precisely as it 
was, or produce such an unimportant alteration in it 
as no one would think of takiiig into account. 

Or again; although the different sets of statistics 
may not as above give almost identical results, yet they 
may do what practically comes to very much the same 
thing, that is, arrange themselves into a small number 
of groups, all of the statistics in a group coinciding 
in their results. If for example a consumptive man 
desired to insure his life, there would be a marked 
difference in the statistics according as we took his 
peculiar state of health into account or not. We 
should here have two sets of statistics, natural kinds 
they might almost be called, which would offer de« 
cidedly different results. If we were to specialize 
still further, by taking into account insignificant 
qualities like those mentioned in the last paragraph, 
we might indeed get more limited sets of statistics 
applicable to persons more closely resembling the in- 
dividual in question, but these would not differ 
sufficiently in their results to make it worth our while 
to do so. In other words, the different series which 



5.] LOGIC OF CHAKCBU 231 

are applicable arrange themselves into a limited 
number of groups, whence the range of choice amongst 
them is very much diminished in practice. 

§ i. It may serve to make the foregoing remarks 
clearer to express them under a slightly different form. 
It was shewn in the first two chapters that, in the 
enquiries to which Probability introduces us, we are 
concerned with a series or iodefinitely extensive class 
which is fixed by the presence of permanent attributes, 
the individuals of it being differenced (and thence a 
sub-class created) by the presence or absence of certain 
variable attributes. The conditions mentioned above 
are equivalent to asserting that these classes must 
be easily distinguishable; in other words, since the 
class is distinguished by means of certain attributes, 
these attributes must either be confined to it, or, if 
they are found elsewhere, must exist there in easily 
distinguishable degrees or in different combinations. 

§ 5. The reasons for the conditions above described 
are not difficult to detect Where these conditions 
exist the process of selecting a series or class to which 
to refer any individual is very simple, and the selection 
is final. The process is simple, for there being but 
a few classes, and these defined by easily distinguishable 
attributes, all we have to do in any particular case is 
to ascertain whether these attributes exist, which ought 
not in general to offer any difficulty. The selection 
also is final; for though the individual possesses many 



232 IXMHC OF CHAVCE. [cHAP. X. 

other attributes which, or the statistics appropiate to 
which, we may gradaallj come to recognize, these will 
not affect the result to any appreciable degree. It is 
assumed, as aboye described, that the consideration of 
these minor qualities does not materially disturb the 
statistics. We do not therefore trouble ourselves 
about their existence when we have onoe determined 
the statistics by which we mean to judge. In any 
case of insurance, for example, the question we have 
to decide is of the very simple kind; Is A. B. a man 
of a certain agel 11 so one in ten like him die in 
the course of the year. If any further questions have 
to be decided they would be of the following description. 
Is A, B, a healthy man? Does he follow a dangerous 
trade 1 But here too the classes in question are but 
few, and the limits by which they are bounded are 
tolerably precise; hence the reference of an individual 
to them is easy. And when we have once chosen our 
class we remain untroubled by any further considera- 
tions ; for since no other statistics are supposed to offer 
a materially different average, we have no occasion 
to take account of any other properties than those 
already noticed. 

§ 6. Let us now examine how isix the above con 
ditions are fulfilled in the case of problems which 
discuss what is called the credibility of testimony. 
The following would be a fidr specimen of one of the 
elementary enquiries out of which these problems are 



SECT. 7.] LOGIC OF CHANGE. 233 

composed; — Here is a statement made hy a witness 
who lies once in ten times, what am I to conclude 
about its truth) Objections might fairly be raised 
against the possibility of thus assigning a man his 
place upon a graduated scale of mendacity. This 
however we will pass oyer, and will assume that the 
witness goes about the world bearing stamped on his 
face the degree of credit to which he has a claim« 
Bat there are other and even stronger reasons against 
the admissibility of this class of problems. 

§ 7. That which has been described in the 
previous sections as the 'individual' which had to be 
assigned to an appropriate class or series of statistics 
is, of course, in this case, a statement, In the par- 
ticular instance in question the individual is already 
assigned to a class, that namely of statements made 
by a witness of a given degree of veracity, but it is 
clearly optional with us to confine our attention to 
this class in forming our judgment; at least it would 
be optional to do this whenever we were practically 
called on to form such an opinion. But in the case 
of this statement, as in that of the man whose insurance 
we were discussing, there are a multitude of other 
properties observable besides the one which is sup- 
posed to mark the given class. As in the latter there 
are, (besides his age) the place of his birth, the nature 
of his occupation and so on ; so in the former there 
are, (besides its being a statement by a certain kind of 



234 LOGIC OF CHANCEL [CHAP. Z. 

witness) the fact of its being uttered at a certain time 
and place and under certain circumstances. At the 
time the statement is made all these qualities or 
attributes of the statement are present te us and we 
have a right to take into accooat as many of them as 
we please. Now the question to be settled seems to 
be simply this; — ^Are the considerations, which we 
might thus introduce, as immaterial to the result in 
the case of a witness, as the corresponding considenir 
tions are in the case of the insurance of a life) There 
can surely be no hesitation in the reply to such a 
question. We soon know all we can know about 
the prospect of a man's death, and therefore rest con- 
tent with general statistics of mortality; but no one 
who heard a witness speak would think of appealing 
to his figure of veracity. The circumstances under 
which the statement is made instead of being in- 
significant are of overwhelming importance The 
appearance of the witness, the tone of his voice, the 
fact of his having objects to gain, together with a 
countless multitude of other considerations which 
would gradually come to light, would make any 
sensible man utterly discard the assigned average. 
He would, in fact, no more think of judging in this 
way than he would of appealing to the Northampton 
tables of mortality to determine the length of life of 
a soldier who was already in the midst of a battle. 
§ 8. It calinot be replied that under these circum^ 



1 



81BGT. 8.] LOGIC OF CHANCE. 235 

stances we still refer the witness to a olass, and judge 
of his veracity hy an average of a more limited kind; 
that we infer, for example, that of men who look and 
act like him under such circumstances, a much larger 
propoHion, say nine-tenths, are found to lie. There 
is no appeal to a class in this way at all, there is no 
immediate reference to statistics of any kind whatever. 
The entire decision seems to depend upon the quicks 
ness of the observer's senses aud of his apprehension 
generally. Statistics about the veracity of witnesses 
seem in &xst to be permanently as inappropriate, am all 
other statistics occasiiMMilly may be. We may know 
accurately the percentage of recoveries after amputa- 
tion of the leg; but what surgeon would think of 
forming his judgment solely by such tables when he 
had a case before him ? I do not deny, of course, that 
the opinion he mighfc form about the patient's pros-> 
pects of recovery might ultimately rest upon the pro-* 
portion of deaths and recoveries he might have pre- 
viously witnessed. But if this were the case, these 
data are lying, as one may say, obscurely in the back- 
ground. He does not appeal to them directly and im- 
mediately in forming his judgment. There has been 
a hx more important intermediate process of appre- 
hension and estimation of what is essential to the case 
and what is not. Sharp senses, memory, judgment, 
and practical sagacity have had to be called into play, 
and there is not therefore the same direct conscious 



236 LOGIC OP CHAKCB. [CHAP. X. 

and sole appeal to statistics that tbere was before^ 
The surgeon may have in his mind two or three in- 
stances in which the operation performed was equally 
severe, but in which the patient's constitution was 
different; the latter element therefore has to be pro* 
perly allowed for. There may be other instances in 
which the constitution was similar, but the operation 
more severe; and so on. Hence, although the ultl* 
mate appeal may be to the statistics, it is not so di- 
rectly; their value has to be estimated through the 
hazy medium of our judgment and memory, which 
places them under a very different aspects 

§ 9. The reader will have a good popular illustra- 
tion of the nature of the difficulty which we have been 
considering, if he will recall to mind any dispute 
which he may have heard or taken part in, in which 
there was an appeal made to the analogy of cases 
similar to that in dispute. Suppose it were the var 
in America. A thinks that the North will win he- 
cause the party which is numerically inferior generally 
loses. (The appeal here, it should be observed, though 
not precisely statistical, is still roughly so : the fidlure 
occurs 'generally;' in Probability it would be properly 
assigned in a numerical proportion. But it is an ap- 
peal of a fundamentally similar character, and the na- 
ture of the argument from it is the same.) B retorts 
that a numerically inferior party when spread over a 
vast country generally is not beaten. A urges that 



i 



SECT. 10.] LOGIC or CHANCE. 237 

slavery is generally a cause of weakness ; not when 
there is a good feeling between the slaves and their 
masters, answers B. And so on ad injmitum. The 
reason why no settlement can thus be come to, is, I 
apprehend, the one given above. There is not here 
any system of natural classification universally recog- 
nized, and appealed to as final, so that there may be 
general agreement as to the class by the statistics ap* 
propriate to which each party is ready to stand or &11. 
The particular ciroumstanoes of the case which may 
£rom time to time come into notice, are here of ex-> 
treme importance from the marked alterations which 
they produce upon the averages. No one could get 
up such a dispute if the question were whether a 
coming child were likely to be a boy or girl. 

§ 10. A criticism somewhat resembling the above 
has been given by Mr Mill {Logic^ Bk. iii. Chap, xviii. 
§ 3) upon the applicability of the theory of Probability 
to the credibility of witnesses. But he has added 
other reasons which do not appear to me to be quite 
valid; he says ''common sense would dictate that it 
is impossible to strike a general average of the vera- 
city, and other qualifications for true testimony, of 
mankind or any class of them; and if it were possible, 
such an average would be no guide, the credibility of 
almost every witness being either below or above the 
average." The latter objection would apply with equal 
force to estimating the length of a man's life from 



238 LOGIC OP CHANCE. [cHAP. 3C 

tables of mortality; for the credibility of different 
witnesses can scarcely have a wider range of variation 
than the length of different lives. If statistics of cre- 
dibility conld be obtained, they would furmsh us in 
the long run with as accurate inferences as any other 
statistics^ the individual variations of excess and de- 
fect being at length neutralised. The original statis* 
tics would however be neglected, because there are 
circumstances in each individual statement which refer 
it most evidently to some new class depending on dif- 
ferent statistics, which latter afford a far better chance 
of being right in that particular case. In most ia* 
stances of the kind in question, indeed, such a change 
is thus produced in the mode of formation of our 
opinion, that the mental operation ceases to be in any 
sense founded on a direct appeal to statistics. Another 
reason moreover for discarding the theory of Probabi- 
lity in these examples is the much greater importance 
of attaining not merely average truth, but truth in 
each instance; we had rather not form an opinion at 
all, than form one which shall only be right in the 
long run. 

The reasons given above seem to me conclusive 
against the propriety of making testimony and its cre- 
dibility subjects of Probability. It is not denied, of 
course, that we may if we please propose questions in 
this form, and then solve them by the theory. When 
we do so however^ since opinions about real witnesses 



8BCT, 11.] liOGIC OP CHANCE. 239 

are not stated or answered in such a way, I can only 
regard tbe expression, ' a witness who lies once in ten 
times,' as being a sort of synonym for ' a bag yielding 
a black ball once in ten times,' introduced into the 
work for the sake of some variety. There are no 
restrictions whatever upon the right of inventiug ex- 
amples for the sake of illustration, or upon the lan- 
guage in which we may express them. We might, if 
we chose, assume that two geometrical theorems fail 
once in ten times, and then determine the chance of a 
solution being correct, which they both agree in sup- 
porting. This would do as well as the example about 
witnesses, at least as an exercise in arithmetic. But 
if we were to propose it as a rule for practical guid-> 
ance, exceptions might begin to be made. 

§ 11. There is however a slightly different view 
of the question which may be taken, and which we 
must now pause for a moment to examine. Because 
decisions as to the truth of any individual statement 
cannot reasonably be regarded as belonging to Proba- 
bility^ it does not follow that this science is inappli- 
cable to help us when we have to decide about the 
truth of a small number of statements. This slight 
change in the nature and extent of the opinion to be 
formed does not indeed make any difference in any of 
the common examples drawn from games of chance, 
for there the appeal is equally to the statistics^ or 
knowledge of the average, whether we are deciding 



240 LOGIC OF CHANCE. [CHAP. X. 

about an iBdividual instance or about several ; and in 
each of these cases alike we have onlj these averages 
to appeal to. But in other applications of Probabi- 
lity the case is slightly altered. It has been already 
pointed out that the individual characteristics of any 
sick man's disease would prevent the surgeon from 
judging of his recovery by statistics alone and directly; 
but if an opinion had to.be formed about a small 
number of cases, in a hospital say, statistics and all 
the inferences they can yield might reasonably be in- 
troduced. The ground of this difference is obvious. 
It arises from the fact that the characteristics of the 
individual, which made us forsake our original average, 
do not produce the same disturbance when. we Have 
to judge about a group of cases. The original average 
stni remains the most available ground on which to 
form an opinion, and therefore Probability again be* 
comes applicable. 

Now it is conceivable that similar results might 
follow if we were to alter the range of our observation 
in the application of Probability to testimony. Be- 
cause no one ought to judge of the truth of a single 
statement by the rules of that science, does it follow 
that he ought not thus to judge of the truth of a suc- 
cession of statements 1 or, what belongs to the same 
class of enquiries, that he should not make use of Pro* 
bability in deciding upon the correctness of the verdict 
9f a juiy ? 



1 



SECT. 11.] LOGIC OP CHANGE, 241 

It must be admitted that this application of the 
theory is not by any means so objectionable as that 
previously discussed. The individual characteristics of 
the statement of the witness, or of the circumstances 
under which it was uttered, which forced us to aban- 
don all appeal to the average of his statements, would 
not exist in like manner amongst all the witnesses or 
all the jury; Hence if we could only find out what 
was the comparative frequency with which true and 
false statements or judgments were delivered, it would 
not be ,so gross an abandonment of the best available 
sources of information if we were to resolve, in any 
individual instance, to adhere to this average in the 
formation of our opinion. Still in the majority of in- 
stances, perhaps in all of them, we might find abund- 
antly sufficient data in each separate instance to make 
an appeal to the average of very inferior value. Poli- 
tical passion, class prejudices, local sympathies, and a 
multitude of other disturbing agencies of this kind, 
could generally be detected in such amount as to pre- 
vent any one who was not strongly biassed towards 
statistics from trusting to the mere averages that 
might be given to him. 



IG 



CHAPTER XL 

ON THE CAUSES BY WHICH THE PECULIAR 
SERIES OF PROBABII/ITT ARE PRODUCED. 



§ 1. The characteristic feature of the phenomena to 
^hich the theory of Probability is applicable, as these 
phenomena present themselves when the njles are 
capable of being immediately applied to them, was 
fully discussed and illustrated in some of the earlier 
chapters. We will now enter into a short examina- 
tion of the causes by which this feature is produced, 
taking up the enquiry at the ppint at which it was 
left in the brief discussion devoted to the subject in 
the ninth chapter. 

§ 2. To divide the sum total of these causes into 
objects and agencies, is to make a division which, 
without pretending to absolute philosophical accuracy, 
will be sufficiently complete for our present purpose. 
In the tossing up of a penny, for example, the objects 
would be the penny or pence which were successively 
thrown ; the agencies, the act of throwing, and eveiy 
thing which combined with this to make any parti- 
cular face turn uppermost. This is a very simple and 
intelligible division, and can easily be extended in 



8ECT. 3.] LOGIC OF OHAKCB. 243 

meaning, I ihmk, so as to embrace evefry class of ob- 
jects with whic^ Probability is concerned. First then 
let us suppose a succession of such objects absolutely 
alike, and let them be exposed to agencies in all re*- 
i^cts identical. We should expect to find this iden- 
tity of antecedents fcdlowed by a similar identity of 
eonsequents. If Mmilar pence, or ihe same penny, 
were thrown in the same way we should expect to 
find the Same face always fall uppermost. 

§ 3. But liow suppose that instead of actual iden- 
tity in the antecedents we had a general uniformity 
with individual rariations in l^e objects, and assume 
that thero is a similar degree of uniformity in the in- 
fioence of the agencies to which these objects are sub- 
jected; it will readily be understood that we may still 
find in most cases* a uniformity of a similar descrip- 
tion in the oooisequenta. 

Wh^i we come to examine the antecedents in the 
examples which experience presents to us, this uni- 
formity will be found almost invariably to exist both 
in the objects themselves, and in the agencies to which 

* I saj * in most cases,' because, as every student of the na- 
tural sciences is well aware, there are instances in which a veiy 
slight difference between the objects, or the agencies to which 
they are exposed, may bring about a vety in^nstderable difference 
in the result. A difference even of degree cadj, — for example, 
temperature, — may cause one of kind, — the qualities of water 
and ice. But these instances are exceptional ; as a general rule 
m-e msy assttme that the resemblance will be perpetuated. 

16—2 



244 LOGIC OP CHANCE. [CHAP. XI* 

they are exposed. In the case of the objects large 
classes will be observed throughout all the individual 
members of which a general resemblance extends. In 
that of the agencies we shall, it is true, perceive an 
extreme perplexity. Analysis will show them to be 
made up of an almost infinite number of different 
components, but it will detect the' same peculiarity, 
that we have so often had occasion to refer to, pervad*- 
ing almost all these components. The proportions in 
which they are combined will be found to be nearly, 
though not qidte, the same; the intensity with which 
they act will be nearly, though not quite, equal. And 
all these uniformities will unite and blend into a more 
perfect harmony, according as we take the average of 
a larger «number of instances. 

Take^ for example, the length of life. According 
to what we know about the subject, the constitutions of 
a very large number of persons selected at random will 
be found to present much the same feature; — ^general 
uniformity accompanied by individual irregularity. 
Now when these persons go out into the world, they 
are exposed to a variety of agencies, the collective in- 
fluence of which will assign to each the length of life 
allotted him. These agencies are very numerous^ — 
climate, food, clothing, &c. — ^together with many others^ 
the nature of which is at present buried in obscurity. 
But^ owing to their adjustment^ the result is that 
these causes do, in a sort of way, balance each other. 



SECT. 5.] LOGIC OP CHANCE. 245 

and produce somethiDg of uniformity. Each becomes 
in its turn a cause, is interwoven inextricably with an 
indefinite number of other causes, and the same kind 
of uniformity is in this way propagated, amidst endless 
individual variations, throughout all nature. 

§ 4. It may be said that this statement is no 
answer to the question with which we started, for 
instead of explaining how a certain state of things is 
caused, it points out that the same state exists else- 
where. There is a uniformity in the objects when 
they are submitted to calculation; we then grope 
about amongst the causes of them, and after all only 
discover a precisely similar uniformity existing amongst 
those causes. This is perfectly true, and indeed 
nothing else could have been fairly expected. Ulti- 
mately no cause cart be assigned, because there is 
none to assign; we can only state it as an experi- 
mental fact that such Is actually the arrangement of 
things. Mediately, however, we can do something, by 
pointing out the means through which the derivative 
results are obtained. Taking this as the object of our 
enquiry, we have seen, by the above analysis, that the 
uniformity in question may be principally assigned to 
two causes or rather conditions. 

§ 5. (I.) There are classes of objects, each class 
containing a multitude of individuals more or less 
resembling one another. Suppose that the phenomenon 
under consideration is the length of life» The objects 



2461 LOGIC OP CHiosrcai [chap, xl 

m thiB caae^ are tke humi^n beings whose lives we ai^ 
CQHsidmng. The rosemhla^ce enlisting amongst ihem 
is to be fouad m the strength and soundness of the 
organs which thej possess, together with all the cir-* 
cumstances whidb collectively make up what we call 
the goodness of their constitntiona The uniformity 
that we may trace in the results is owing, much more 
than is often suspected, to this arrangement of things 
in natural kinds, each kind containing a large number 
of individuals^ Were each kind of animals limited to 
a single pair, or even, to but a few pairs, thevo would 
not be much scope left for the collectiou of statistical 
tables amongst them. Or, to make a less violent 
supposition; — If the numbers iu each naturaL cla^s of 
objects were much smaller than they sbre at present, 
and the distinctions between the classes somewhati 
more marked^ the consequent inapplicability of any 
kind of statistical tables to them^ though not quite 
fiital, would still be yfixjf sedjous. A large number of 
objects in the. dass^ togetheir with that general simir 
larity which entitles t^e objects to be fairly com- 
prised in one class, seea» to be importa&t coiaditions for 
the applicability of the theory of Probability to ajaj' 
phenomena. 

§ 6. (II.) By ti>e a<^usitment of the relative, in- 
tensity of the different forces and agencies in nature 
and the respective frequency of their occurrence, the 
effects which these produce are alsa tolerably uniform. 



SKCT. 7.] LOGIC OF CHANGE. 247 

It ia quite coaceivable that this second oondition 
should correct the formep by converting this general 
unifcHTBiitj into an absolute one, or that on the other 
hand it should aggravate it into utter want of 
uniforniitT. As a matter of fact this second condition 
does neither, but simplj varies the details, leaving 
the uniformity offredsely the same general description 
as it was before. Of, if the objects are supposed to 
be absolutely alike, as in* the case of successive throws 
of a penny when the same penny is always thrown, it 
may serve to create this kind of uniformity. Thus, 
to recur to a former instance; One man overworks 
himself another ibllows an unhealthy trade, a third 
exposes himself to iBfeetion, &ei, ; the result of all this 
isr that the length of mien's lives> like the strength 
of their cosuitijtutions^ preserves, when tabulated, a 
tolacabl« regularity. 

§ 7. Thei reader must observe tfeixi^t this oondition is 
arbitrary, in. the same sense in which the- fibrmep con- 
dition was arbitrary, and in a greater degree. We 
not m^ely can coneeive the absence of such uniformity 
in thje agencies; we can easily find instances m which 
uniformity is actually wanting- Thus the length of life 
is tolerably regular, and so aore the numbers' who die 
in successive years, or centuries of most of the common 
complaints. But is it so of aU diseases? What of the 
Sweating sicknessi, the Black deaths the Asiatie cholera? 
I am not denying that these events have their causes^ 



248 LOGIC OF CHANCE. [cHAP. XI. 

and tliat they would be produced again by the re- 
currence of the conditions which caused them before. 
But they do not reciu*; at least not the former diseases. 
They seem to have depended upon such rare conditions 
that their occurrence was almost solitary; and when 
they did occur their course was so excentric and ir- 
regular as to entirely deprive their results (that is, 
the number of deaths . which they caused) of the 
statistical uniformity of which we are speaking. 

We can only lay it down therefore as a general^ not 
a universal rule, that the agencies in question show 
the kind of uniformity which is requisite to make the 
objects affected by them fitting subjects of Probability. 
It may be replied that the occasional irregularity just 
alluded to only arises from the fact of our having con- 
fined ourselves to too limited a time, and that we 
shall see it disappear here, as elsewhere, if we keep 
our tables open long enough. This reply is conclusiTe 
only upon the supposition that the ways and thoughts 
of men are, in the long run, invariable, or subject only 
to periodic changes. On the assumptioa of a steady 
progress in society either for the better or the worse, 
the argument falls to the ground at once. From what 
we know of the course of the world these fearful pests 
of the past may be considered as solitary events in 
our history. Or at least events which will not be re- 
peated. No uniformity would therefore be found in 
the deaths which they occasion, though the registrar's 



SECT. 8.] IiOGIC OP CHANCEL 249 

books should be kept open for a million years, and 
these agencies are therefore for the most part ex- 
cluded from the science of Probability. 

§ 8. Having thus examined the process by which 
the results in question are brought about, it may now 
be interesting to spend a short time in the enquiry, 
What are the principal classes of things amongst 
which such conditions are to be discovered? 

(I.) These conditions prevail principally, I ap- 
prehend, in the properties of natural kinds; both in 
the ultimate and in the accidental and derivative 
properties. In all the characteristics of natural 
species; — in all they do, and in all which happens to 
them, so far as it depends upon their properties — we 
sddom fail to discover this regularity. Thus in men, 
their height, strength, weight; the age to which they 
live, the deaths of which they die; all present a well- 
known uniformity. Life Insurance tables offer a 
good instance of the multiplicity and importance of 
the above-mentioned applications of Probability. 

(II.) The same peculiarity prevails in the force 
and frequency of most natural agencies. Winds and 
storms are seen to lose their proverbial irregularity 
when examined on a large scale. Man's work there- 
fore when operated on by such agencies as these, even 
though it had been made in different cases absolutely 
alike to begin with, afterwards shows only a general 
regularity. I may sow exactly the same amount of 



n 



250 LOGIC OF CHAJTCB. [CHAP. XI. 

seed in my field eveiy year. One season tlie yield 
may be moderate, the next be extraordinarily abun^ 
dant through hot dry weather, and the thiixl be mnch. 
injured by hail. But Ib the long nm these irregulari- 
ties will disappear in the result of my crops, beeause 
they disappear in the power and frequency of the 
productive agencies. The businesa of underwriters. 
Fire Insurance, &e.y will fiill pnncipaUy under this 
head, though in some respects they are more connected 
with the former. Bat the distinction which is thus 
made, is, after all,, principally one of arrangement;, 
these natui:al agencies are closely assimilated to the- 
propeiiiies of natural kinds, and indeed might perhapa 
be considered such,, if we were to extend our observar- 
tions. 

§ &. The above are instances of natural objects 
and natuval agenciea I am inclined to believe that, 
it is, in such only, as distinguished frpm< things artificial, 
that the propei*ty in question is to be found; This is 
an asseHion that will need some discussion, and expla* 
nation. Two instances, in apparent opposition, will at 
once occur to the mind of some readers, one of which 
from its great intrinsie impoi'tance, and the other from. 
the frequency of the problems which it furnishes, will 
demand sepai-ate examination. 

§ 10. (1) In the course of observation, by as- 
tronomical and (Kther instruments, the utmost pessible 
degree of accuracy is offcen desired, & degree which 



SECT. 10.] LOGIC OF CHAKCE. 251 

cannot be attained by any one single observation. 
What we do therefore in these cases is to make a very 
large number of different observations, which are 
naturally found to differ somewhat from one another 
in their results ; by means of these the true value is 
to be found as accurately as possible. This process is 
one which astronomers have such constant occasion to 
perform that a special rule (that known as the rule of 
least squares) has been invented for the purpose. I 
have already alluded to this rule in a former chapter, 
and need only say at this point that its object is to 
determine the unknown true result from a considerable 
number of the known but slighty incorrect results. 

The subjects then of calculation here are a cei^n 
number of elements — slightly incorrect elements — 
given by successive observations. Are not then 
these observations artificial, or the direct product of 
voluntary agency? I think not; at least it rather 
depends on what we understand by voluntary. What 
is really intended and aimed at by the observer is, of 
course, perfect accuracy, that is, the true observation, 
or the voluntary steps and preliminaries on which 
this observation depends. Whether voluntary or 
not this result only can be called intentional. Bat 
this result is not obtained. What we actually get in 
its place is a series of deviations from it, containing 
results more or less wide of the truth. Now by what 
are these deviations caused? It appears to me that 



252 LOGIC OF CHANCE. [CHAP. XI. 

agencies are at work here, similar for the most part 
to those whose operation we have ju&t been considering 
in some of the previous sections. Heat, dust, friction, 
draughts of air, are some of the causes which divert 
us from the truth. Besides these there are other 
causes which certainly depend upon human agency, 
but which are not strictly speaking voluntary; on 
the contrary they owe the character they possess, 
of general unifoimity only as opposed to absolute 
uniformity, to the fact of their being involuntary. 
They are such as the irregular action of the muscles, 
inability to make our organs execute the precise 
purposes we have in mind, &c. All these conditions, 
though utterly incalculable singly, vary in the long 
run tolerably uniformly. 

§ 1 1. A few words may here be added to the re- 
marks made in the second chapter upon this rule of 
Least Squares. It would be presumptuous in me to 
attempt any criticism of the mathematics themselves 
by which the rule is supported or proved, I only wish 
to make it plain why I cannot regard the mathematics 
as necessarily rigidly appropiiate. The possibility of 
determining the correct observation, by the help of 
any formula, when the actual data before us are a 
series of slightly incon-ect observations, seems to imply 
that these incorrect elements will group themselves in 
some determinate and orderly way about the correct 
one. And the possibility of determining the requisite 



SECT. 12.] LOGIC OF CHAKGB. 253 

formula d priori implies that the law according to 
which the elements thus group themselves must be 
the same or similar under all possible different circum- 
8tanoe& It was urged in a former chapter, that if this 
9tate of things did really exist, it would seem that a 
great extent of special experience would be necessary 
to verify it. Kow that we have made a brief exami- 
nation of the different agencies by which any uniform* 
ity of the kind in question is generally produced in 
the cases in which it does actually exist, there seems 
still more reason for such an appeal to experience. 
The series to which Probability is applied aro generally 
the result of a combination of many and complicated 
agencies, and these agencies often show a tendency, 
under certain circumstances, so to change their effects 
that the rules of Probability may at length become 
inappropriate. Unless then the series of observations 
to which the rule of Least Squares is applied are en« 
tii-ely unlike most of the other natural seiies in which 
Probability is made use of (a conclusion which I have 
attempted to disprove above), we surely ought to insist 
upon something more than the purely abstract d priori 
principles which are commonly offered in support of 
the rule. 

§ 12. (2) The other example to which I allude is 
the stock one of cards and dice. Here, as in the last 
case, the result is remotely voluntary, in the sense 
that it would not be produced at all but for human 



256 LOGIC OF CHAHCE. [cHAP. XI^ 

imiformity which is not to be found in a single design. 
So far as this is the case, however, it would be a re- 
turn to the principles laid down in the opening of this 
chapter. The height which the different builders con- 
templated might be found to group themselyes into 
something of the same kind of uniformity as that 
which prevails in most other things which they should 
undertake to do independently. We might then trace 
the action of the same two conditions; — ^a harmony in 
the multitude of their different designs, a harmony 
also in the infinite variety of the influences which 
have modified those designs. But this is a very dif- 
ferent thing from saying that the work of one man 
will show such a result as this. The difference is 
much like that between the tread of a thousand men 
who are stepping without thinking of each other, and 
their tread when they are drilled into a regiment. In 
the former case there is the working of a thousand 
minds, in the latter of one only. The former therefore 
would introduce us to the province of Probability, the 
other would not. 

§ 15. The foregoing very brief enquiry into the 
causes by which the peculiar form of statistical results, 
with which we have been throughout concerned, are 
actually produced, must suffice here. Any fuller 
discussion would seem to belong more properly to a 
fiir wider science, in fact to the general philosophy of 
Inductive evidence. The conditions upon which the 



1 



SECT. 15.] LOGIC OP CHANCE. 257 

production of our general statistical propositions de- 
pends, as distinguished from the inferences to be made 
from them when they are obtained, lie outside the 
confines of the science of Probability. 



17 



CHAPTER XII. 

FALLACIES. 

§ 1. In works on Logic a chapter is generally 
devoted to the discussion of Fallacies, that is, to the 
description and classification of the difi'erent ways in 
which the rules of Logic may be transgressed. The 
analogy of Probability to Logic is sufficiently close to 
make it advisable to adopt the same plan here. In 
describing my own opinions I have been, of course, 
often obliged to describe and criticize those of others 
when they seemed erroneous. But some of the most 
widely spread errors find no supporters worth mention- 
ing, and exist only in vague popular misapprehension. 
It°will be found the best arrangement, I think, at the 
risk of occasional repetition, to collect and classify a 
few of the errors that occur most frequently, and as 
far as possible to trace them to their sources. In 
doing so I shall for the most part confine myself to 
the special province of this work, the application, 
namely, of Probability to moral and social science, 
and shall avoid the discussion of isolated problems in 
games of chance and skill except when some error of 
principle seems to be involved in them. 



SECT. 2.] LOGIC OP CHANCE. 259 

§ 2. (I.) One of the most fertile sources of error 
and coDfusion upon the subject has been already 
scTeral times alluded to, and in part discussed in a 
previous chapter. This consists in choosing the class 
to which to refer an event, and therefore judging of 
the rarity of the event and the consequent improba- 
bility of foretelling it, after it has Jiappened, and then 
transferring the impressions we experience to a sup- 
posed contemplation of the event beforehand. No 
error need arise in this way if we were careful as to 
the class which we thus selected; but such carefulness 
is often neglected. 

An illustration may serve to make this plain. A 
man once pointed to a small target chalked upon a 
door, the target having a bullet hole through the 
centre of it, and surprised some spectators by declaring 
that he had fired that shot from an old fowling-piece 
at a distance of a hundred yards. His statement was 
true enough, but he suppressed a rather important 
fact. The shot had really been aimed in a general 
way at the bam door, and had hit it; the target was 
afterwards chalked round the spot where the bullet 
struck. A deception analogous to this is, I think, 
often practised unconsciously in other matters. "We 
judge of events on a similar principle, feeling and 
expressing surprise in an equally unreasonable way, 
and deciding as to their occurrence on grounds which 
are really merely a subsequent adjunct of our own. 

17—2 



/ 



1 



260 LOGIC OF CHANCE. [CHAP. XII. 

Butler's remarks about the story of Csesar, discussed 
already in the fifth chapter, are of this character. 
He selects a series of events from history, and then 
imagines a person guessing these correctly who at the 
time has not the history before him. As I have 
already pointed out, it is one thing to be unlikely to 
guess an event rightly without specific evidence; it is 
another and very different thing to judge of the truth 
of a story which was founded upon evidence. But it 
is a great mistake to transfer to one of these ways of 
viewing the matter the mental impressions which 
properly belong to the other. It is like drawing the 
target afterwards, and then being surprised that the 
shot lies in the centre of it. 

§ 3. One aspect of this fallacy has been already 
discussed, but it will serve to clear up difficulties 
which are often felt upon the subject if we re- 
examine the question under a somewhat more general 

form. 

In the class of examples under discussion we are 
generally presented with an individual which is not 
indeed definitely referred to a class, but in regard to 
which we have generally no difficulty in choosing the 
appropriate class. Now suppose we were contemplat- 
ing such an event as the throwing of sixes with a pair 
of dice four times running. Such a throw would be 
termed a very unlikely event, and the odds against 
its happening would be said to be 36x36x36x.36-l 



SECT. 5.] LOGIC OF OHAKCE. 261 

to 1 or 1679615 to 1. The meaning of these phrases, 
as has been abundantly pointed out, is simply that 
the event in question occurs very rarely; stated 
with numerical accuracy, it occurs once in 1679616 
times. 

§ 4. But now let us make the assumption that 
the throw has actually occurred; let us put ourselves 
into the position of contemplating sixes four times 
running, when it is supposed to be known or reported 
that this throw has happened. The same phrase, 
namely that the event is a very unlikely one, will 
often be used in relation to it, but we shall find that 
this phrase introduces extremely different meanings. 
Properly speaking Probability is scarcely applicable 
then ; the throw in question being supposed to have 
happened, things are in a stage in relation to that 
throw in which all inferences from the science of Pro- 
bability are superseded. The event is known, and 
therefore we need not now judge by means of statis- 
tics as to what might have been expected to occur. 
When however, as is' often the case. Probability is 
appealed to in reference to such a throw, we shall 
find that two or three quite distinct meanings are in- 
troduced. 

§ 5. (1) There is, firstly, the most correct mean- 
ing. The event, it is true, has happened, and we 
know what it is, and therefore, as just stated, we have 
not really any occasion to resort to the rules of Pro^ 



262 LOGIC OF CHANCE. [cHAP. XH. 

bability; but we can nevertheless conceive ourselves 
as being in the position of a person who does not 
know, and who has only Probability to appeal to. 
By calling the chances 1679615 to 1 against the 
throw we then mean to state the fact, that inasmuch 
as such a throw occurs only once in 1679616 times, 
our guess, were we to guess, would be correct 
only once in the same number of times, that is 
if it were a fair guess simply on these statistical 
grounds. 

§ 6. (2) But there is a second and very different 
conception sometimes introduced, especially when the 
event in question is supposed to be known, not as 
above by the evidence of our experience, but by the 
report of a witness. "We may then mean by the 
* chances against the event' (as was pointed out in 
Chapter v.) not the proportional number of times we 
should be right in guessing the event, but the pro- 
portional number of times the witness will be right 
in reporting it. The grounds of our inference are 
here shifted altogether. In the former case the 
statistics were the throws and their respective fre- 
quency, now they are the witnesses' statements and 
their respective truthfulness. 

§ 7. (3) But there is yet another meaning some- 
times intended to be conveyed when persons talk of 
the chances against such an event as the throw in 
question. They may mean — not, Here is an event. 



fiBOr. 7.] LOGIC OF CHANCE. 263 

how oflen should I have guessed it ] — nor. Here is a 
report, how often will it be correct 1 — ^but something 
entirely different from either, namely, Here is an 
event, how oflen will it be found to be produced by 
some one particular cause ? 

This meaning will often be found to introduce 
itself in the case of coincidences. When, for example, 
a man hears of dice giving as above the same throw 
several times running, and speaks of this as very ex- 
traordinary, we shall often find that he is not merely 
thinking of the improbability of his guess being rights 
or the report being true, but, along with this, of the 
throw having been produced by fair dice*. There is, 
of course, no reason whatever why such a question as 
this should not be referred to Probability, provided 
always that we could find the appropriate statistics by 
which to judge. These statistics would be composed, 
not of throws of the particular dice, nor of reports of 
the particular witness, but of the occasions on which 
such a throw as the one in question respectively had 
and had not been produced fairly. The objection to 
this view of the question would be that no such sta- 
tistics are obtainable, and if they were, we should 
prefer to form our opinion (on principles described in 
Chapter ix.) from the special circumstances of the 
case rather than from an appeal to the average. 

* There are some remarks on this in Mill's Logic, Bk. ni. 
ch. XXV. 



264 LOGIC OF CHANCE. [CHAP. XIL 

§ 8. The readei* will easily be able to supply any 
ntuuber of examples in illustration of the distinctions 
just given; we will briefly examine but one. I bide 
a banknote in a certain book in a large library, and 
leave the room. A person tells me that after I went 
out a stranger came in, walked straight up to that 
particular book, and took it away with him. Many 
people on hearing this account would reply. How 
extremely improbable ! On analysing the phrase, I 
think we shall find that certainly two, and possibly 
all three, of the above meanings are involved in this 
exclamation. (1) What may be meant is this, — As- 
suming that the report is true, and the stranger inno- 
cent, a rare event has occurred. Many books might 
have been thus taken without that particular one 
being selected. I should not therefore have expected 
the event, and when it has happened I am surprised. 
Now a man has a perfect right to be surprised if he 
pleases, but he has no logical right {on this view) to 
make his surprise a ground for disbelieving the event 
To do this is to fall into the fedlacy described at the 
commencement of this chapter. The fact of my not 
having been likely to have guessed a thing beforehand 
is no reason for doubting it when I am told of it. 
(2) Or I may stop short of the events reported, and 
apply the rules of Probability to the report itself If 
so, what I mean is, as has been several times described, 
such a story as this now before me is of a kind veiy 



SECT. 9.] LOGIC OF CHANCE. 205 

generally false, and I cannot therefore attach much 
credit to it now. (3) Or I may accept the truth of 
the report, but doubt the fact of the stranger having 
taken the book at random. If so, what I mean is 
that of men who take books in the way described, only 
a small proportion will be found to have taken them 
really at random; the majority will do so because 
they had by some means found out what there was 
inside the book. 

§ 9. Each of the above three meanings is a pos- 
sible and a legitimate meaning. The only requisite 
is that we should be very careful to ascertain which of 
them is present to the mind, so as to select the appro- 
priate statistics. The first makes in itself the most 
legitimate use of Probability ; the only drawback being 
that at the precise time in question the functions of 
Probability are superseded by the event being other- 
wise known*. The second or third, therefore, is the 
more likely meaning to be present to the mind, for in 
these cases Probability, if it could be practically made 
use of, would, at the time in question, be a means of 
drawing really important inferences. The drawbacks 
here, are the impossibility of finding such statistics, 
and the extreme disturbing influence upon these sta- 

* The fallacy described at the commencement of this Chapter 
consists in failing to observe this, and appealing to the statis- 
tics appropriate to the first meaning to draw a oouclusion which 
ought to rest on those appropriate to the second or third. 



LOGIC OP CHANCE. [CHAP. Xfli 

tistics of the circumstances of the special case. Al- 
though, therefore, we frequently draw conclusions in 
«uch a case on the principles of the science, we cannot 
do this with any such approach to accuracy as would 
justify us in obtaining numerical results. 

§ 10. (11.) The fallacy described at the com- 
mencement of this chapter arose from determining to 
judge of an observed or reported event by the rules of 
Probability, but employing an entirely wrong set of 
statistics in the process of judging. Another fallacy, 
closely connected with this, arises from the practice of 
taking some only of the characteristics of such an 
event, and arbitrarily confining to these the appeal to 
Probability. An example may serve to make this 
plain. I toss up twelve pence, and find that eleven of 
them give heads. Many persons on witnessing this 
would experience a feeling which they would express 
by the remark, How near that was to getting all 
heads ! And if any thing very important were staked 
on the throw they would be much excited at the oc- 
currence. But in what sense were we near to twelve ? 
The number eleven of course is nearer to twelve than 
ten or nine are, but there is surely something more 
than this in the person's mind at the moment. There 
is a not uncommon error, I apprehend, which consists 
in unconsciously regarding the eleven heads as a thing 
which is already somehow secured so that one might, 
as it were, keep them and then take our chance for 



SECT. 11.] LOGIC OP CHANCE, 267 

the odd one. The eleven are mentally set aside, looked 
upon as certain (for they have already happened) and 
we then introduce the notion of chance merely for the 
twelfth. But this twelfth, having also happened, has 
no better claim to such a distinction than any of the 
others. If we will introduce the notion of chance in 
the case of the one that gave tail we must do the 
same in the case of all the others. In other words, if 
the tosser be dissatisfied at the appearance of the one 
tail, and wish to cancel it and try his luck again, he 
must toss up the whole lot of pence again fairly to- 
gether. In this case, of course, so far from his having 
a better prospect for the next throw he may think 
himself in very good luck if he makes again as good 
a throw as the one he rejected. 

§ 11. In the above example the error is so trans- 
parent that a very slight amount of reflection will 
enable any one to see through it. But in forming a 
judgment upon matters of greater complexity than 
dice and pence, especially in the case of what are 
called 'narrow escapes,' a mistake of an analogous 
kind is, I apprehend, far from uncommon. A person, 
for example, who has just experienced a narrow escape 
will often be filled with surprise and anxiety amount- 
ing almost to terror. The event being past, these 
feelings are, at the time, in strictness inappropriate. 
If, as is quite possible, they are merely instinctive, or 
the result of association, they do not fall within the 



268 LOGIC OP CHANCE. [CHAP. XII. 

province of any kind of Logic. If however, aa seems 
to me far more likely, they arise from a supposed 
transference of ourselves into that point of past time 
at which the event was just about to happen, and the 
production by ima^nation of the feelings we should 
then expect to experience, this process partakes of the 
nature of an inference, and can be right or wrong. 
In other words, the alarm may be proportionate or 
disproportionate to the amount of danger that might 
fairly have been reckoned upon in such a hypothetical 
anticipation. If we attend to the remarks people 
make on such occasions, we shall find, I think, that 
they do distinctly consider that their feelings admit of 
justification; if so, I do not perceive by what other 
process they can be justified than by that which has 
been just described. If the supposed transfer were 
completely carried out, there would be no fallacy ; but 
it is often very incompletely done, some of the com- 
ponent parts of the event being supposed to be deter- 
mined or 'arranged' (to use a sporting phrase) in the 
form in which we now know that they actually have 
happened, and only the remaining ones being fairly 
contemplated as future chances by Probability. 

A man, for example, is out with a friend, whose 
rifle goes off by accident, and the bullet passes 
through his hat. He trembles with anxiety at think- 
ing what might have happened, and perhaps remarks^ 
*How very near I was to being killed!' Now we 



5ECT. 11.] LOGIC OF CHANCE. 269 

may safely assume that be means sometliing more than 
that a shot passed very close to him ; such an event 
might have been produced purposely and cautiously 
by his friend, and in that case his feelings would have 
been totally different. He has now some vague idea 
that, as he would probably say, ^ his chance of being 
killed then was very great.* His surprise and terror 
may be in great part physical and instinctive, arising 
simply from the knowledge that the shot had passed 
very near him. But his mental state may be analysed, 
and I think we shall find, at bottom, a fallacy of the 
kind described above. To speak or think of chance 
in connection with the incident, or to refer to what 
might have been, is to refer the particular incident to 
a class of incidents of a similar character, and then to 
consider the comparative frequency with which the 
contemplated result ensues. Now the series which 
we may suppose to be most naturally selected in this 
case is one composed of shooting excursions with his 
friend; up to this point the proceedings are assumed 
to be designed, beyond it only, in the subsequent 
event, was there accident. Once in a million times 
perhaps on such occasions the gun will go off acci- 
dentally; one in a thousand only of those discharges 
will be directed near his friend's head. If we will 
make the accident a matter of Probability, we ought 
by rights in this way (to adopt the language of the 
first example), to ' toss up again ' fairly. But we do 



270 LOGIC OP CHANCE. [CHAP. XII. 

not do this ; we seem to assume for certain that the 
shot goes within an inch of our heads, detach that 
from the notion of chance at all, and then begin to 
introduce this notion again for possible deflections 
from that saving inch. In such a case one's prospects 
naturally become very disagreeable. 

§ 12. If the reader will try to analyse his feelings 
just after he has had a narrow escape himself, or wit- 
nessed one in others, I think he will find that this 
fallacy is generally to some extent involved in them. 
The mere proximity to danger cannot be the cause of 
the anxiety, for in other cases where the danger was 
equally near, but from which the notion of chance is 
excluded, no such anxiety is felt. ,1 do not think 
that any one can make any justification of his feelings, 
or would naturally attempt to make one, without in- 
troducing the notion of chance. We shall find it 
scarcely possible to explain what we feel without in- 
troducing an "if," or putting ourselves mentally into 
the same position again, and then thinking about the 
different issues of the event which might be expected 
as a general rule under those circumstances. If in 
such a position the probability of danger would be 
really great, terror would not have been inappropriate 
then, and therefore anxiety is a very legitimate pro- 
duct of imagination afterwards. But if, on the other 
hand, as is very often the case when all the chances 
are contemplated, even that amount of proximity to 



SECT. 13.] LOGIC OF CHANCE. 271 

danger which was actually experienced was extremely 
improbable, then no justification can be offered for 
the subsequent anxiety. 

§ 13. (III.) A common mistake is to assume that 
a very unlikely thing will not happen at all. It is a 
mistake which, when thus stated in words, is too ob- 
vious to be committed, for the meaning of an unlikely 
thing is one that happens at rare intervals ; if it were 
not certain that the event would happen at rare in- 
tervals it would not be called unlikely. This is an 
error that could only occur in vague popular misap- 
prehension, and is so abundantly refuted in other 
works on Probability, that I shall touch upon it very 
briefly here. It follows of course, from our definition 
of Probability, that to speak of a very rare combina- 
tion of events as one that is * sure never to happen,' is 
to use language incorrectly. Such a phrase may pass 
as popular exaggeration, but otherwise it is either 
tautological or contradictory. The truth about such 
rare events cannot be better described than in the 
following quotation from De Morgan*: — 

'^ It is said that no person ever does arrive at such 
extremely improbable cases as the one just cited [draw- 
ing the same ball five times running out of a bag con- 
taining twenty balls]. That a given individual should 
never throw an ace twelve times running on a single 
die, is by fai* the most likely; indeed, so remote are 

* Essay on Probabilities, p. 1 26. 



272 LOGIC OP CHANCE. [CHAP. Xtl. 

the cliaiices of such an event in any twelve trials (more 
than 2,000,000,000 to 1 against it) that it is unlikely 
the experience of any given country, in any given 
century, should fiimish it. But let us stop for a mo- 
ment, and ask ourselves to what this argument applies. 
A person who rarely touches dice will hardly believe 
that doublets sometimes occur three times rimning ; 
one who handles them frequently knows that such is 
sometimes the fact. Every very practised user of those 
implements has seen still rarer sequences. Now sup- 
pose that a society of persons had thrown the dice so 
often as to secure a run of six aces observed and re- 
corded, the preceding argument would still be used 
against twelve. And if another society had practised 
long enough to see twelve aces following each other, 
they might still employ the same method of doubting 
* as to a run of twenty-four, and so on, ad infinitwni. 
The power of imagining cases which contain long com- 
binations so much exceeds that of exhibiting and ar- 
i*anging them, that it is easy to assign a telegraph 
which should make a separate signal for every grain of 
sand in a globe as large as the visible universe, upon 
the hypothesis of the most space-penetrating astrono- 
mer. The fallacy of the preceding objection lies in 
supposing events in number beyond our experience, 
composed entirely of sequences such as fall within our 
experience. It makes the past necessarily contain the 
whole, as to the quality of its components ; and judges 



PECT. 14.] LOGIC OP CHANCE. 273 

by samples. Kow the least cautious buyer of grain 
requires to examine a handful before he judges of a 
bushel, and a bushel before he judges of a load. But 
relatLvely to such enormous numbers of combinations 
as are frequently proposed, our experience does not 
deserve the title of a handful as compared with a 
bushel, or even of a single grain*." 

§ 14. The origin of this inveterate mistake is not 
difficult to be accounted for. It arises, no doabt^ from 
the exigencies of our practical life. No man can bear 
in mind every contingency to which he may be ex- 
posed. If therefore we are ever to do anything at all 
in the world, a large number of the rarer contingencies 
m.ast be entirely left out of account. And the neces-' 
sity of this oblivion is strengthened by the shortness 
of our life. Mathematically speaking it would be said 
to be certain that any one who lives long enough will 
be bitten by a mad dog, for the event is not an impos- 
sible^ but only an improbable one, and must therefore 
come to pass in time. But this and an indefinite 
number of other disagreeable contingencies have on 
most occasions to be entirely ignored in practice, and 
thence they come almost necessarily to drop equally 
out of our thought and expectation. And when the 
event is one in itself of no importance, like a rare 
throw of the dice, it requires almost an eflfort of imagi- 

* Euay <m Proboibilities, p. 126, 

18 



274 LOGIC OP CHANCE. [6HAP. XII. 

nation to some persons to realise the throw as being 
even possible. 

§ 15. There is one particular form of this error 
which, from the importance attached to it by some 
writers, deserves perhaps more special examination. 
As stated above, there can be no doubt that, however 
unlikely an event may be, if we (loosely speaking) vary 
the circumstances sufficiently, or if, in other words, we 
keep on trying long enough, we shall meet with such 
an event at last. K we toss up a pair of dice a few 
times we shall get doublets; if we try longer with 
three we shall get triplets, and so on. However un- 
usual the event may be, even were it sixes a thousand 
times running, it will come some time or other if we 
have only patience and vitality enough. Now apply 
this result to the letters of the alphabet. Suppose 
that one at a time is drawn from a bag which contains 
them all, and is then replaced. If the letters were 
written down one after another as they occurred, it 
would commonly be expected that they would be 
found to make mere nonsense, and would never 
arrange themselves into the words of any language 
known to men. No more they would in general, but 
it is a commonly accepted result of the theory, and 
one which we may assume the reader to be ready to 
admit without further discussion, that, if the process 
were continued long enough, words making sense 
would appear; nay more, that any book we chose to 



SECT. 16.] LOGIC OP CHANCE. 275 

mention, — Milton or Shakespeare, for example, — 
would be produced in this w&j at last. It might take 
more years than ve have space in this volume to 
represent in figures to obtain such works, but come 
they would at last. Now many people have not 
unnaturally thought it derogatory to genius to 
suggest that its productions could have also been 
obtained by chance, whilst others have gone on to 
argue, If this be the case, might not the world itself 
in this manner have been produced by chance? 

§ 16. *Dugald Stewart makes a reference to 
some remarks of Condorcet upon this subject, and has 
thought his reasonings of sufficient importance to need 
a detailed criticism. I will only remark here on 
this particular point, that any such inference as this 
about the creation of the world, involves in addition 
the fallacy described in § 9. It confounds together 
the probability of our foretelling an event with the 
probability of the event having been produced in 
some given way. But the other inference, — ^that, I 
mean, about the production of a Shakespeare, — seems 
equally startling and capable of leading to identical 
conclusions, whilst its meaning is unmistakable, and 
its truth not likely to be disputed by students of Pro- 
bability. 

It may console some persons to be reminded that 

♦ Dugald Stewart's Works, edited by Sir W. Hamilton, 
Vol.711, p. 115. 

18—2 



276 Xioaic op chance. [chap. xn. 

the power of producing a Shakespeare, in time, is not 
confined to consummate genius and to mere chance. 
Any one, down almost to an idiot, might do it, if 
he took sufficient time about the task. For suppose 
that the required number of letters were arranged, 
not by chance but designedly, and according to the 
rules of the theory of permutations : their number 
being really finite, every order in which they could 
occur would come in its due turn, and therefore 
every thing which can be expressed in language 
would be arrived at some time or other, the works of 
Shakespeare of course amongst other things.. It 
would probably take about as long to do it one way 
as the other, but with unlimited time either plan 
would be feasible. 

§ 17. There is really nothing that need startle or 
shock anyone in such a theory. It arises from the follow- 
ing cause. The number of letters, and therefore of 
words at our disposal is limited; when therefore any* 
thing is to be expressed in language it necessarily 
becomes subject to this limitation. The possible 
variations of thought are literally infinite, so are those 
of spoken language (by intonation of the voice, <fec.); 
but when we come to words there is a limitation 
which is distinctly conceivable by the mind, though 
the restriction is one that in practice will never be 
appreciable. The answer therefore is plain, and it is 
one that will apply to many other cases as well, that 



SECT. 18.] IiOGIC OF CHANCE. 277 

to put a finite limit upon the number of ways in 
wliich a tbiDg can be done is to determine that any 
one who is able and willing to try long enough shall 
succeed in doing it. If a great genius condescends to 
do it under these circumstances, he must submit to 
the possibility of having his claims disputed by the 
chance-man or idiot. If Shakespeare were limited 
to ten words, the time within which the latter agents 
might claim equality with him need not be very 
great. As it is, having the range of the English 
language at his disposal, his reputation is not in any 
present danger of being assailed on such grounds. 

As an additional security it may be remarked 
that each of these latter agents, even when in course 
of time they had stumbled upon a brilliant conception, 
would probably not be aware of the fact, so that prac- 
tically they would require a Shakespeare at their 
elbow to tell them at which of their performances 
they had better stop. 

§ 18. (IV.) In discussing the nature of the 
connexion between Probability and Induction, we 
examined the claims of a rule commonly given for 
inferring the probability that an event which had 
been repeatedly observed would recur again. I en- 
deavoured to show that all attempts to obtain and 
prove such a rule were necessarily futile; if these 
reasons were conclusive the employment of such a 
rule must of course be regarded as flEdlacious, There 



278 LOGIC OF CHANCE. [CHAP. XII. 

is no necessity to repeat here the arguments which 
were employed on a former occasion; I will only re- 
call the reader's attention to the following considerar* 
tions, which were then but very briefly touched upon. 

Instead of there being one single rule of succession 
we might divide the possible forms of the rule into 
three classes. 

§ 19. (1) In some cases when a thing has been 
observed to happen several times it becomes in con- 
sequence more likely that the thing should happen 
again. This agrees with the ordinary form of the 
rule, and is probably the case of most frequent occur- 
rence. The necessary vagueness of expression when 
we talk of the 'happening of a thing' makes it quite 
impossible to tolerate the rule in this general form, 
but if we specialize it a little we shall find it assume 
a more familiar shape. If, for example, we have ob- 
served two or more properties to be fiwjuently as- 
sociated together in a succession of individuals, we 
shall conclude with some force that they will be found 
to be so connected in future. The strength of our 
conviction however will depend not merely on the 
number of observed coincidences, but on far more 
complicated considerations; for a discussion of which 
the reader must be referred to regular treatises on 
Inductive evidence. Or again, if we have observed 
one of two events succeed another several times, the 
occurrence of the former will excite in most cases 



SECT, 21.] LOGIC OF CHANCE. 279 

some degree of expectation of the latter. As before^ 
liowever, the degree of our expectation is not to be 
assigned by any simple formula; it will depend in part 
upon the supposed intimacy with which the events 
are connected. This would lead to a discussion upon 
laws of causation, and the circumstances under which 
their existence may be inferred. 

§ 20. (2) Or, secondly, the past recurrence may 
in itself give no valid grounds for inference about the 
future; this is the case which most properly belongs 
to Probability. That it does so belong will be easily 
seen if we bear in mind the fundamental conception 
of the science. "We are there introduced to a series, — 
for purposes of inference an indefinitely extended 
series, — of terms, about the details of which informa- 
tion is not given; our knowledge being confined to 
the statistical fact, that, say, one in ten of them has 
some attribute which we will call X. Suppose now 
that Hve of these terms in succession have been X, 
what hint does this give about the sixth being also an 
XI Clearly none at all; this past fact tells us no- 
thing; the formula for our inference is still precisely 
what it was before, that one in ten being X it is 
one to nine that the next term is X. And however 
many terms in succession had been of one kind^ 
precisely the same formula would still be given. 

§ 21. The way in which events will justify the 
answer given by this formula is often misunderstood. 



280 LOGIC OP CHANCE. [CHAP. XIE 

Por tlie benefit therefore of those unacquainted with 
some of the conceptions familiar to mathematicians^ I 
vill add a few words of explanation. Suppose then 
that we have had X twelve times in successioii* 
This is clearly an anomalous state of things. To 
suppose anything like it to continue for ever would 
be obviously in opposition to the statistics, which 
assert that in the long run only one in ten is X. 
But how is this anomaly got over? In other words, 
how do we obviate the conclusion that X's will occur 
more frequently than once in ten times, after such a 
long succession of them as we have now had 9 Many 
people seem to believe that there must be a diminution 
of X's afterwards to counterbalance their past pre- 
ponderanca This however would be quite a mistake; 
the proportion in which they occur in future must 
remain the same throughout; it cannot be altered if 
we adhere to our statistical formula. The fact is that 
the rectification of the exceptional disturbanoe in the 
proportion will be brought about simply by^we con- 
tinual influx of fresh terms in the series. Th4ii4 ^^^ 
in the long run neutralize the disturbance, not by 
any special adaptation, as it were, for the purpose^ 
but by the mere weight of their overwhelming numbera 
At every stage therefore, in the succession, whatever 
might have been the number and natui*e of the pre- 
ceding terms, it will still be true to say that one in 
.ten of the tenPB will be an X« 



fiECT. 21.] LOGIC OF CHANCE. 281 

If we had to do only vith a finite number of 
terms, however large that number might be, such a 
disturbance as we have spoken of would, it is true, 
need a special alteration in the subsequent propor- 
tions to neutralize its effects. But when we have 
to do with an infinite number of terms, <this is not 
the case ; the 'limit' of the series, which is what 
we then have to deal with (Ch. iii. § 33), is entirely 
unaffected by these temporary disturbances. In the 
continued evolution of the series we shall find, as a 
matter of fact, more and more of such disturbances, 
and these of a more and more exceptional character. 
But whatever the point we may occupy at any time, 
if we look forward or backward into the indefinite 
extension of the series, we shall still see that the 
ultimate limit to the proportion in which its terms 
are arranged remains the same; and it is with this 
limit, as above mentioned, that we are concerned in 
the strict rules of Probability. Suppose, for illus- 
tration, that there is one part in a thousand of salt 
in sea-water. Put some fresh water in a cistern; then 
however much there may be of the fresh, if only salt 
water enough is turned on, the only proportion that 
will be ultimately approximated to will be the same 
as that in the sea itself, viz. one part in a thousand of 
salt. To effect this it will not be necessary to suppose 
the subsequent sea- water to be more salt; the pro- 
,portion will be restored, or, to speak strictly, will 



282 LOGIC OF CHANCE. [CHAP. XtL 

tend to be restored simply by adding for ever water of 
the original degree of saltness. 

The commonest example, perhaps, of this kind is 
that of tossing up a penny. Suppose we have had four 
heads in succession; people have tolerably realized 
by now that 'head the fifth time' is still an even 
chance, as 'head' was each time before, and will be 
ever after. The preceding paragraph explains how 
it is that these occasional disturbances in the average 
become neutralized in the long run. 

§ 22. (3) There are other cases which, though, 
rare, are by no means unknown, in which such an 
inference as that obtained from the Bule of Succession 
would be the direct reverse of the truth. The oftener 
a thing happens, it may be, the more unlikely it is to 
happen again. This is the case whenever we are 
drawing things from a limited source (as balls from a 
bag), or whenever the act of repetition itself tends to 
prevent the succession (as in giving false alarms). 

I am quite ready to admit that we believe the re- 
sults described in the last two classes on the strength, 
of some such general Inductive rule, or rather prin- 
ciple, as that involved in the first. But it would be 
a great error to confound this with an admission of 
the universal validity of the rule in each special in- 
stance. We are speaking about the application of 
the rule to individual cases, or classes of cases; this is 
quite a distinct thing, as was pointed out in a previous 



«£CT. 23.] LOGIC OF CHAKCB. 283 

chapter, from giving the grounds on which we rest 
the rule itselfl If a man were to lay it down as a 
uniyersal rule^ that the testimony of all persons was 
to be believed, and we adduced an instance of a man 
having lied, it would not be considered that he saved 
his rule by shewing that we believed that it was a lie 
on the word of other persons. But it is perfectly 
consistent to give as a general (not a universal) rule, 
that the testimony of men is credible, then to separate 
off a second class of men whose word is not to be 
trusted, and finally, if any one wants to know our 
ground for the second rule, to rest it upon the first. 
If we were speaking of necessa/n/ laws, such a conflict 
as this would be as hopeless as the old 'Cretan' 
dilemma; but in instances of Inductive and Analogical 
extension it is perfectly harmless. 

§ 23. A familiar example, about which many 
people must have disputed at one time or another of 
their lives, will serve to bring out the three different 
possible conclusions mentioned above. We have 
observed it rain on ten successive days. A and B 
conclude respectively for and against rain on the 
eleventh day; G maintains that the past rain affords 
no data whatever for an opinion. Which is right? 
We really cannot determine ^ priori. An appeal must 
be made to direct observation, or means must be 
found for deciding on independent grounds to which 
class we are to refer the instance. If, for example, it 



284 LOGIC OP CHANCE. [cHAP. XII* 

•were known that every country produces its own 
rain, we should choose the third rule, for it woidd be a 
case of drawing from a limited supply. If again we 
had reasons to believe that the rain for our country 
might be produced anywhere on the globe, we should 
probably conclude that the past rainfall threw no 
light whatever on the prospect of a continuance of 
wet weather, and therefore take the second. Or ii^ 
finally, we knew that rain came in long spells or 
seasons, as in the tropics, to have had ten wet days 
in succession would make us believe that we had 
entered on one of these seasons, and that therefore 
the next day would probably resemble the preceding 
ten. 

Since then all these forms of such an Inductive 
rule are possible, and we have often no ct priori 
grounds for preferring one to another, it would seem 
to be unreasonable to attempt to establish any 
universal formula of anticipation. All that we can 
do is to ascertain what are the circumstances under 
which one or other of these rules is, as a matter of 
feet, found to be applicable, and to make use of it 
under those circumstances. 



CHAPTER XIII. 

ON THE CREDIBILITY OF EXTRAORDINARY 
STORIES. 

§ 1, It is now time to recur for fuller investigation 
to an enquiry which has been already briefly touched 
upon more than once; I mean the validity of testimony 
to establish, as it would be said, an otherwise im- 
probable story. It will be remembered that in a 
previous chapter (the fifth) we devoted some ex- 
amination to an assertion of Bishop Butler, which 
seemed to be to some extent countenanced by Mr 
Mill, that a great improbability before the event 
might become but a very small improbability after 
the event. In opposition to this I endeavoured to 
^how that the different estimate which we undoubtedly 
formed of the credibility of the examples adduced, 
had nothing to do with the fact of the event being 
past or futui'e, but arose from a very different cause ; 
that the conception of the event which we entertain 
at the moment (which is all that is then and tl^ere 
actually present to us, and as to the correctness of 
which as an adequate representation of facts we have 
to make up our minds) comes before us in very 
different ways. In one case it was a mere guess of 



286 LOGIC OP CHANCE. [cHAP. XHI, 

our own which we knew from statistics would be 
right in a certain proportion of cases; in the other it 
was the assertion of a witness, and therefore the 
appeal was not now to statistics of the event, but to 
the trustworthiness of the witness. The conception, 
or 'event' if we will so term it, had in fact passed out 
of the category of guesses (on statistical grounds) into 
that of assertions (most likely resting on some specific 
evidence), and would therefore be naturally regarded 
in a veiy different light. 

§ 2. But it may seem as if this principle would 
lead us to far more startling conclusions than any 
which we reject. For, by transferring the appeal 
from the frequency with which the event occurs to 
the trustworthiness of the witness who makes the 
assertion, is it not implied that the probability or im- 
probability of an assertion depends solely upon the 
veracity of the witness? If so, ought not any story 
whatever to be believed when it is asserted by a 
truthful person? 

Undoubtedly it ought, on the data now before us. 
But let it be clearly understood what conditions are 
implied in this limitation. Only under the two 
following conditions is it true that the credit we give 
to the statement of a witness is entirely independent 
of anything in the nature of the event to which he 
testifies. In the first place, the question must be 
really one of Probability ; that is^ the asserted event 



SECT. 3.] LOGIC OP CHANCE. 287 

must be only rare, or in other words, mnst be ad- 
mitted actually to happen sometimes, though we may 
not know the frequency of its occurrence. In the 
second place we must adhere strictly to our data, and 
judge of the trustworthiness of the witness in any 
particular case simply from the statistical frequency 
with which he tells truth and falsehood. 

We will now briefly examine the meaning of these 
conditions, and ascertain the consequences when tliey 
are not adhered to. We shall thus be able to clear 
lip a great deal of confusion which seems to exist 
upon this subject. It will also appear how very 
narrow is the province of pure Probability, and how 
arbitrary therefore are the restrictions which have 
to be introduced if we will insist upon judging of 
ordinary events by its rules. 

§ 3. We will begin with the second of the above- 
mentioned conditions. In judging of the probability 
of any assertion of a witness of * given veracity, it is 

* I have already (Chap, x.) giyen reasons against the pro- 
priety of applying the rules of Probability with any strictness to 
such examples as these. But, although all approach to nume- 
rical accuracy is unattainable, we do undoubtedly recognize a 
distinction in ordinary life between the credibility of one wit- 
ness and another ; such a rough practical distinction as this will 
be quite sufficient for the purposes of this Chapter. For con« 
Tenience, and to illustrate the theory, the examples will mostly 
be stated in a numerical form, but it is not intended to be im- 
plied that any such accuracy is really attainable in practice. 



288 LOGIC OF OHANCB. [CHAP. XIII. 

implied that our sole ground for attributing truth to 
the story is that a given proportion of the assertions 
made by that witness are found to be true. In other 
words, it is implied that the particular inference, 
namely the truth of the statement in question, is 
simply a deduction from the general proposition, 
namely, the proportion ofthe statements of the witness 
which are true. 

§ 4. The meaning and propriety of such an im« 
plication will appear more clearly if we examine what 
follows from neglecting or denying it. This will be 
best introduced by the examination of a phrase which 
is often employed in these discussions, and which the 
reader must have frequently met, I mean ' a balance 
or contest of opposite improbabilities.' It was Hume, 
I believe, in his Essay on Miracles, who first brought 
the expression into general use, but it has been 
adopted by Paley and by most of the other opponents 
of Hume, and has met with very general acceptance. 
What is meant by such a phrase, I apprehend, is 
this; — ^that in forming a judgment upon the truth of 
any assertion, we find that the assertion is comprised 
in two or more different classes, and that according 
as we referred it to one or other of these different 
classes our judgment as to its truth would be different. 
In other words, there are two or more different sources 
of evidence, any one of which, if we adhered to that 
alone, would lead us to a conclusion at variance with 



SECT. 5.] LOGIC OF CHANCS. 289 

that whicli is supported by the others. It seems to 
be assumed in the phrase, 'contest of opposite im- 
probabilities/ that, when these different sources of 
evidence coexist together, they would all in some way 
retain their probative force so as to produce a contest, 
ending generally in a victoiy to one or other of them. 
I should say, on the contrary, that, in the case under 
discussion, if we adhere to our data one of these 
sources of evidence simply and entirely supersedes 
the other; if we do not adhere to the data there is 
a contest, no doubt, but it is one for the decision of 
which no rule can be given, and the result of which is 
therefore entirely arbitrary. 

§ 5. We adhere to our data if we judge of the 
truth of the individual assertion entirely from the 
frequency with which the witness speaks truth and 
falsehood If so, of course, one of the sources of 
evidence entirely supersedes the other. It is clearly 
intended when the data are given to us that this 
should be the case. For by the improbability of the 
event is meant nothing else than the improbability of 
our guessing it, owing to the rarity of its occurrence. 
This is one possible ground on which our opinion 
might have been rested. But the witness is not sup- 
posed to have guessed, or, if he did guess, this, like 
any other source of error on his part, is already in- 
cluded in the figure which describes his veracity. 
This is quite a different ground on which to rest our 

19 



290 LOGIC OF CHANGE. [CHAP. XIII. 

conviction, and when this is given it clearly supersedes 
the other. Any datum therefore which involves the 
veracity of a witness carries its paramount authority 
on its face, for when this is given all necessity for us 
to guess is removed. 

§ 6. The course described in the last section is 
the one which ought to be taken^ and in that case all 
'contest' is evaded. But when a contest is provoked 
the result is very different. Suppose that a witness 
who lies once in ten times tells a very extraordinary 
story. When the statement is made our datum is, 
as before, This is a statement made by a witness who 
lies but once in ten times; and from this only ought 
we to draw our inference But practically we seldom 
submit to this restraint. Some snch reflection as the 
following will almost unavoidably arise; — This state- 
ment is of a suspicious kind, that is, it is of a kind 
which, when made by people generally, is in most cases 
found to be false. Kow if we take these extraneous 
considerations into account, what conclusions are we 
to draw? Here there is a real contest of the kind 
already discussed in the chapter on Induction (chap. 
VII. § 15), and I do not see how any possible way 
of deciding the contest can be found, except by in- 
troducing an arbitrary supposition, or by adding to 
our information by a fresh appeal to specific expe- 
rience. 

§ 7. Our perplexity is as follows; — We have 



SECT. 8.] LOGIC OP CHANCaa. 291 

brfore us a statement. On this occasion it is made 
by a witness who lies in the long run but once in ten 
times; it is however a statement of a kind which is 
generally fiilse; stated numerically it is found (let us 
suppose), when we examine the case of many different 
witnesses, to be false ninety-nine times in a hundred. 
We are here brought to a dead lock, our science 
offering no principle by which we can form an opinion 
or attempt to decide the question. It was shown in 
the chapter on Induction, already referred to, that 
an indefinite number of conclusions were all equally 
possible. For example, all the witness's extraordinary 
assertions might be true, or they might all be false, 
or they might be true and false in almost any propor- 
tion whatever. This being the case, how can any 
definite rule be obtained for the solution of the dif- 
ficulty? The issue of the contest when, as in this case, 
there is a contest, seems hopelessly indeterminate. 

§ 8. I am quite aware that some solutions have 
been offered. Hume speaks of our deducting one pro- 
bability fi:om the other and apportioning our belief 
to the remainder. Doubtless he would have laid no 
stress upon the numerical accuracy of the process, but 
even this semblance of accuracy must be abandoned 
Archbishop Thomson considers that one probability 
entirely supersedes the other. Were he confining his 
remarks to the class of instances at present under 
notice, this would be correct, but a reference to his 

19—2 



292 LOGIC OP CHANcaB. [chap, xhl 

remarks in the Louws of Thc/ught will show, I think, 
that this is not the case. 

There is one conclusive objection to all such solu- 
tions of the difficulty; they are all attempts to decide 
d priori what can only be decided by a specific appeal 
to experience. It has been maintained throughout, 
in this Essay, that any rule for apportioning the 
amount of our belief must, if the rule claims to be 
correct, contain a correct assertion about the statistical 
.frequency of events. If therefore it can be shown (as 
I shall now attempt to show) that the events in 
question are perfectly indeterminate, such a rule is at 
once condemned. 

§ 9. The way in which the error creeps in is 
as follows; — The witness is said to report an 'im- 
probable event;' it is inferred therefore that his 
veracity must in that case be more questionable. X 
dislike speaking of an improbable event in Prober 
bility, though one is often obliged to do so, but let it 
be clearly understood what is meant by the term. It 
denotes simply an event which does happen, bat 
happens rarely, and of the existence of which therefore 
we should, judging only by statistics, be extremely 
doubtful But why should a witness be at all lees 
likely to tell the truth when relating a rare event 
than when he is relating a common event? It may- 
be that he is, but it is also possible that the reverse 
should be the case; if so, a moment's consideration 



SECT. 10.] LOGIC OP CHANCE. 293 

will show that it is only by the supposed effect of the 
rarity of the event upon the vemcity of the witness 
that his story becomes more or less probable. 

§ 10. For suppose that the fact of a witness 
describing a very uncommon event makes him more 
than usually careful, and therefore actually adds to 
his veracity. "We should in that case receive his ex- 
traordinary assertions with even more readiness than 
his ordinary assertions. This is in reality no such 
unlikely supposition. Let us assume that a man of 
ordinary intelligence and a philosopher — ^say Professor 
Owen^ — make some assertion about common things; 
we believe them both. Let them now each describe 
some extraordinary Itisiis naturce or monstrosity 
which they report that they have seen. I presume 
that almost every one would believe the assertion of 
the Professor nearly as readily in the latter case as in 
the former, whereas when the same story came from 
the uninstructed man it would be received with great 
hesitation. Whence arises the difference? From the 
assumption that the philosopher's assertions will be to 
the full as accurate in matters of this kind as in thosd 
of the most ordinary description, whilst in the case of 
the other man we are far from feeling this confidence. 
Even if the reader is not prepared to go this length 
he will allow, I presume, that the difference of credit 
which he would attach to the two assertions, when 
coming from the philosopher, would be very much less 



294 LOGIC OF CHANCE. [CHAP. Zm. 

than vhat it would be when they came from other 
men; the admission of any such difference, no matter 
to what extent, is an admission of the principle here 
contended for. 

§ 11. Instances of the kind jnst mentioned are of 
course exceptional; as a general rule the known com- 
parative rarity of the event asserted does add to the 
improbability of the assertion. It may add to it to 
any amount whatever. There are many forms of 
lying gossip about the fidsehood of which we feel so 
certain, that we should hardly believe them though 
coming from the mouth of the most trustworthy of men. 
Whilst therefore it is quite true, in certain instances, 
that the more uncommon the event the more impro- 
bable does the testimony of the witness become who 
asserts it, it is also true in other instances that the 
direct reverse is the case. Sometimes the rarity of 
the event may make the testimony actually more 
trustworthy; sometimes it may, though taken into 
account, leave the testimoDy unaltered. We do not 
argue directly from the rarity of the event; our belief 
though undoubtedly often influenced by the rarity, 
is only influenced by the supposed effect which that 
may have upon the veracity of the witness. 

I consider therefore that, on the rules of Probabili- 
ty, whenever a story of any kind whatever is described 
as coming from a witness of a given degree of veracity, 
our only course is to accept the story as having that 



SBCT. 13.] LOGIC OP CHANCE. 205 

giirea degree of probability m its favour* If we do 
not do this we £Edl into utter confusion. All attempts 
to allow for the alteration of the witness's veracity ac- 
cording to the description of story which he tells must, 
from the nature of the case, be entirely arbitrary or 
depend upon the sagacity of the observer, — ^no fixed 
rules can help us. 

§ 12. Hitherto we have proceeded on the as- 
sumption that the question is properlj one of PrOf 
bability. In other words, we have assUmed it to be 
admitted that the event asserted to have happened 
does or will really happen, the only claim it has to 
be called improbable being that owing to its rarity 
we should be very little inclined to expect its occur- 
rence in any particidar instance. But such an as- 
sumption cannot long be practically adhered to, for 
the term improbable,' when applied to an event, has 
a far wider signification in popular estimation. We 
will therefore make a brief enquiry into what follows 
when this assumption is denied. It is the more 
necessary to do so, because it is this aspect of the 
question which possesses most popular interest, 
especially in reference to the credibility of testimony 
when applied to miraculous stories. 

§ 13. The best way of approaching the subject 
will be by examining two or three examples in sue? 

* See thd note at t|ie end of this Chapter. 



296 LOGIC OP CHANcae. [chap. xni. 

cession. We shall then perceive that there are two 
distinct principles at work, upon one of which the 
science of Probability is competent to speak, whilst 
upon the other it is not. 

(1) A witness tells me that he has thrown sixes 
five times nmning. This is a question of pure Pro- 
bability; on the grounds already so fully discussed^ 
we form our opinion solely from the degree of trust- 
worthiness of the witness. 

(2) Again,' he tells me that he has seen a sheep 
with five legs. The event which he reports is known 
to happen sometimes, perhaps not more unfrequently 
than the throw of the dice in the last example. If 
the question is proposed to us as one of pure Proba- 
bility, and if the credibility of the witness is our only 
datum, we judge^ as in the last example^ solely by this 
credibility. 

Or, if we do not confine ourselves to our dat% 
we might biing the assertion under the category of 'ex- 
traordinary stories.' We might consider that^ from its 
partaking of the marvellous, the stoiy is one of a kind 
commonly foimd to be erroneous, and therefore change, 
in some arbitrary proportion, the degree of credibility 
we assign to the witness. 

(3) Again, he says that he has seen a sheep 
with ten legs. We feel here that we are getting on 
to different ground. The practical indisposition to 
confine ourselves to our data, which was considerable 



SECT. 14.] LOGIC OP CHANCE. 297 

ia the last example, is now uncontrollable. The ex- 
iBtence of such a monster, as that described, is doubt- 
ed; this doubt, which will intrude itself into the 
question, can only be settled, if it is to be noticed at all, 
by Inductive principles and skill in applying them to 
natural science; Give what weight we will to the 
unwarranted alternative suggested in the last ex- 
ample — in other words, alter as we will the figure 
of the witness's veracity, owing to the fact of his 
story being extraordinary — it will fail to satisfy. 
The doubt we feel is far too serious for its force to 
be spent by a mere arbitrary correction of the vera- 
city. 

§ 14. It is owing in part to considerations such 
as these that we are forced, as it appears to me, to 
take the view of Probability adopted in this work. 
I see no other satisfactory way than to separate the 
province of Probability distinctly from that of Induc- 
tion, to assign to the latter the task of preparing the 
statistics, vouching for their truth, and extending 
them as far as admissible, and to relegate to the 
former the far narrower task of finding rules for in- 
ferring the particular and individual cases from these 
extended statistics. 

It may perhaps be urged in reply, that the case of 
this monstrosity does not differ essentially from many 
examples that we meet with in pure Probability. It 
is true (it may be said) that such a sheep has not yet 



298 LOGIC OP CHANCE. [CHAP. Xm. 

been seen, but after all it may be like some rare col- 
location of dice or balls which will come at last in its 
turn, and is not therefore more unlikely than any 
other particular event of the kind True, it mayj 
and also it may not; and the existence of this possible 
alternative is what necessitates the adoption of further 
evidence. To argue that because it is not more 
difficult to believe in a throw that has not been 
hitherto experienced than in one that has, that there- 
fore it is equally easy to believe in any event hitherto 
unexperienced, is to lose sight of the foundations of 
the science. What we reason from in the case of the 
balls is not the limited series which has actually 
occurred, but the infinite series which analogy and 
Induction lead us to adopt. As shown in Chapter in, 
yre calculate from a substituted series, not from the 
fragment given to us; from potential, therefore, not from 
actual experience. In the case of the balls or dice 
this substitution is so natural that its validity is never 
doubted; moreover, there is a tolerably complete 
unanimity as to the terms which the succession of 
throws can be expected from time to time to produce. 
But in the case of the monstrosity it is very seriously 
doubted whether such a term is in the series at alL 
To say that such a sheep may yet be produced, if we 
wait long enough, is to assert what is possible, but 
more than that is wanted. Before we can calculate the 
chance of the monstrosity by our own guessing, we 



9BCT. 16.] j/>Qic OF CHANCB. 299: 

must know in what proportion of cases things of its 
kind occur. Before we can do so from the witness's 
assertion, we must at least be satisfied that sucL 
creatures are possible, and that the witness has no 
bias towards lying about them; then the knowledge 
of their frequency, as already described, would not be 
needed. But, till these prior questions are settled, 
nothing can be inferred. For the solution of them 
Probability must invoke the aid of Observation and 
Induction, and pause till they have pronounced. 

§ 15. It appears therefore to me that the at- 
tempts so often made to apply the theory of Proba- 
bility, by means of the credibility of testimony, to 
the establishment or otherwise of miraculous stories, 
involve some confusion and ambiguity in the meaning 
of the word 'improbable.' As I have already so 
frequently said, an improbable event in our science is 
one which is admitted to occur sometimes, but whose 
occurrence we should not have anticipated, owing to 
its rarity; popularly it means an event whose occur- 
rence is doubtful, and iS' perhaps the very point in 
dispute. 

§ 16. The foregoing results may be summarized 
as follows; — To have it given, as our only datum, 
that a certain story is asserted by a witness of given 
veracity, binds us down to judging of its truth, what- 
ever may be the nature of the story, solely by the 
relative fi*equency with which the witness speaks 



300 LOGIC OP CHAXCK [CHAP. XIH. 

truth and &,lseliood. Practically, however, we generally 
transgress our data, in which case the problem assumes 
one of two forms. It may remain one of Probability, 
by our admitting that the event asserted does some- 
times happen, but considering that as an 'extra- 
ordinary' event it is not to be judged by the average 
veracity of the witness; then we should generally 
(but not always) diminish the probability of his ve- 
racity, but the amount of this diminution is entirely 
arbitrary. Or the problem may cease to be one of 
Probability by our not admitting that the event does 
or may occasionally happen; then the question must 
be resigned to the far wider science of Induction and 
of evidence generally. 

§ 17. A few words may be added here about the 
combination of testimony, though no new principles 
seem to be required for the discussion of this subject. 
Suppose that two witnesses whose veracity is respect- 
ively ^js and ^ combine in asserting the same story ; 
what strength should this give as to the truth of the 
story? This question seems to me to fall under the 
second of the rules of inference discussed in the fourth 
chapter. The chance of a lie from the witness A is 
^ffj the chance of a lie from B is xV^ therefore the 
chance of a combination of lies from the two is yj^ 
Now, since both A and B agree in making the same 
statement, their statement must be true unless they 
are both lying. The chance therefore that the event 



SECT. 17.] LOGIC OP CHANCE. 301 

has not happened is xJ^y, or the odds in its favour are 
as 119 to 1. 

Practically this combination of testimony generally 
meets ns as a modification of that aspect of the 
question described in the sixth section. A witness, 
A, makes a statement, but it strikes us as being a 
statement of an extraordinary or suspicious character. 
Hence we very much diminish our estimate of his 
veracity for questions of that kind. Then B comes 
forward and confirms the story ; we correct his figure 
of veracity in the same way. But the combination of 
the two statements so greatly increases the probability 
of the truth of the thing stated, as to outweigh in 
many cases the separate diminution of their values. 
The product of two factors, after each has been 
diminished, may be greater than either was singly 
before. Similarly if there be more than two 
witnesses. 

The main practical objection against the value of 
such conclusions as these is the extreme difficulty, in 
£&ct the impossibility in many cases, of being certain 
that the witnesses are independent. The more nearly 
the stories resemble one another the nearer are we to 
the real combination of testimony in question, and 
therefore the greater should be our confidence in the 
truth of what is asserted. But these are precisely the 
cases in which our doubt will be greatest as to the 
witnesses being independent. The nature of the 



302 LOGIC OF CHANGE. [CHAP. XIII^ 

subject seems to forbid any such strict rules as are 
demanded in Logic. 

§ 18. It appears to me therefore that the science 
of Probability has really little or no direct bearing 
upon the credibility of miracles, that is if we limit the 
applicability of the science to that class of enquiries 
to which it seems most properly to belong. Their 
credibility must be established on some independent 
ground before we can judge of the validity of testimony 
to support them. Let it be shown that they are not 
intrinsically incredible, and a combination of testimony 
might establish them as it might establish anything 
else. If we do not take this course, the way in which 
the mere force of testimony might be evaded is obvious 
from what has been already said. We show that ten 
persons agree in supporting a miraculous story; but 
the opponent cannot be prevented from merely put- 
ting his arbitrary alteration of the veracity of the wit- 
nesses at such a point as to make the testimony of ten 
persons when reporting such a thing of very little value. 
The question of miracles has however been so in- 
cessantly introduced in reference to the credibility of 
testimony and thence to the Science of Probability, 
that I cannot forbear devoting a few pages to the 
subject; though, as mentioned above, I conceive the 
connection between miracles and Probability to be 
only indirect. 

§ 19. A necessary preliminary will be to decide 



SBCT. 19.] LOGIC OF CHANCE. 303 

upon some definition of a miracle. It will, I ap- 
prehend, be admitted by most persons that in calling 
a miracle *a suspension of a law of causation,' we are 
giving what^ though it may not amount to an adequate 
definition, is at least true as a description. It is true^ 
though it may not be the whole truth. It is this 
aspect moreover of the miracle which is now exposed 
to the whole brunt of the attack, and in support of 
which therefore the defence has generally been carried 
on. Por these reasons we will commence with this 
view of it. 

Now it is obvious that this, like most other de- 
finitions or descriptions, makes some assumption as to 
matters of fact, and involves something of a theory. 
The assumption clearly is, that laws of causation 
prevail universally, or almost universally, throughout 
nature, so that infractions of them are marked and 
exceptional This assumption is made, but I do not 
think that anything more than this is necessarily 
required. I mean that there is nothing which need 
necessarily restrict us to one or other of the two 
principal schools which are divided as to the nature 
of these laws of causation. The definition will serve 
equally well whether we understand by law nothing 
more than uniformity of antecedent and consequent^ 
or whether we assert that there is some deeper and 
more mysterious tie between the events than mere 
sequence. The term 'causation' in this sense is com- 



304 LOGIC OF CHANCE. [CHAP. XIII. 

men to both schools, though the one might consider it 
inadequate; we may speak, therefore, of 'suspensions 
of causation' without committing ourselves to either. 

§ 20. It should be observed that the aspect 
of the question suggested by this definition is one from 
which we can hardly escape. Attempts indeed have 
been sometimes made to avoid the necessity of any 
assumption as to the universal prevalence of law and 
order in nature, by defining a miracle from a different 
point of view. A miracle has been called, for instance, 
*an immediate exertion of creative power,' *a sign of 
a revelation,' or, still more vaguely, an * extraordinary 
event.' But nothing would be gained by adopting 
any such definitions as these. However they might 
satisfy the theologian, the student of physical science 
would not rest content with them for a moment. He 
would at once assert his own belief, and that of all 
other scientific men, in the existence of universal law, 
and enqxiire what was the connection of our definition 
with this doctrine. An answer would imperatively 
be demanded to the question, Does the miracle, as you 
have described it, imply an infraction of one of these 
laws, or does it not? And an answer must be given, 
unless indeed we reject his assumption by denying 
our belief in the existence of this imiversal law, in 
which case of course we put ourselves out of the pate 
of argument with him. The necessity of having to 
recognize this fact is growing upon men day by day. 



1 



SECT. 21.] LOGIC OP CHANCE. 305 

with the increased study of physical science. And 
since this aspect of the question has to be met some 
time or other it is as well to place it in the front. 
The difficulty, in its scientific form, is of course a 
modem one, for the doctrine out of which it arises is 
modem. But it is only one instance, out of many 
that might be mentioned, in which the growth of 
some philosophical conception ha,s gradually affected^ 
and at last shifted the battle-ground, in some discussion 
with which it might not at first have appeared to 
have any connection whatever. 

§ 21. So far our path is plain. Up to this point 
disciples of very different schools may advance together; 
for in laying down the above doctrine we have care- 
fully abstained from implying or admitting that it 
contains the whole ti'uth. But from this point two 
paths branch out before us, paths as different from each 
other in their character, oiigin, and directicm, as can 
well be conceived. As this chapter is only a digres- 
sion, I will confine myself to stating briefly what 
seem to be the characteristics of each, without attempt- 
ing to give the arguments which might be used in 
their support. 

(I.) On the one hand, we may assume that this 
principle of causation is the ultimate one. By so 
describing it, we do not mean that it is one from which 
we consciously start in our investigations, as we do 
fcom the axioms of geometry, but rather that it is the 

20 



306 LOGIC OF CHANGE. [CHAP. XIII. 

final result towards which we find oarselves drawn by 
a study of natur& Finding that» throughout the 
scope of our enquiries, event follows event in never- 
fitiling uniformity, and finding moreover (some might 
add) that this experience is supported by a tendency 
or law of our nature (it does not matter here how we 
describe it), we may come to regard this as the one 
great principle on which all our enquiries should rest. 

(II.) Or, on the other hand, we may admit a 
class of principles of a very different kind. Allowing 
that there is this uniformity so fiir as our experience 
extends, we may yet admit what I can see no other 
way of describing than by calling it a Superintending 
Providence. To adopt an aptly chosen distinction of 
Professor Kingsley's, it is not to be understood as 
over-ruling events, but rather as tmderlf/ing them. 
I repeat again, that it is not my present object to 
enter into any discussion as to whence we obtain this 
ide& 

§ 22. I cannot see how any reflecting mind can 
fail, at the present time, to be subject to one or other 
of these prepossessions. This word prepossession has, 
in common use, a bad signification almost equivalent 
to prejudice, but I use it here because I know no 
other which would express my meaning equally welL 
If the principles by which we judge were intui- 
tively and immediately obvious, we might conceive 
fuxj one to be able in a sort of way to divest him- 



SECT. 23.] LOQIO OF CHAKCIL 307 

self of prepossessions. These principles could be 
stated, like the axioms on the first pages of Euclid^ 
and anj casual bystander would be able to state in 
two seconds whether he adopted them or not. This 
gives the appearance of commencing absolutely ah 
origine. But if first principles are only to be acquired 
by long and laborious reflection, such a demand as 
this is quite out of the question. To require a dis- 
putant to strip himself of his axioms, flourish them in 
his opponent's face, and then deliberately put them on 
again, is to require him to appeal from his mind when 
mature to what it was when immature. Though ac- 
quired in the most unexceptionable way such prin- 
ciples may, in reference to any particular problem, be 
called prepossessions; and if they are challenged or 
denied, all that any one can do is to repeat his con- 
yiction of their truth, and his belief that other persons 
by reflection will come to regard them as he doesj he 
cannot compel the assent of his opponent. 

§ 23, Now it is quite clear that according as we 
come to the discussion of any pai*ticular miracle or 
extraordinary story under one or other of these pre- 
possessions, the question of its credibility will assume 
a very different aspect. It has been strangely over- 
looked, in many recent discussions, that although a 
diflerence about facts is one of the conditions of a 
bond fide argument, a diflerence which reaches to 
ultimate principles is feital to all argument. The 

20—2 



308 LOGIC OF CHANCE. [cHAP. XHL 

poBsibilitj of present conflict is banished in such a 
case as absolutely as that of future concord. A large 
amount of recent literature on the subject of miracles 
seems to labour under this hopeless defect. Arguments 
have been brought for and against the credibilitj of 
stories without the disputants (on one side at least) 
appearing to have any adequate conception of the 
chasm which separated one side from the other. 

§ 24. The following illustration may serve in 
some degree to show the sort of inconsistency of 
which I am speaking. A sailor reports that in some 
remote coral island of the Pacific, on which he had 
landed by himself, he had found a number of stones 
on the beach disposed in the exact form of a cross. 
Now if we conceive a debate to arise about the truth 
of his story in which it is attempted to decide the 
matter simply by considerations about the validity 
of testimony, without introducing the question of the 
existence of inhabitants, we shall have some notion of 
the unsatisfactory natui*e of many of the current 
arguments about miracles. All illustrations of this 
subject are imperfect, but a case like this, in which a 
supposed trace of human agency is detected interfering 
with the orderly sequence of other natural causes, is 
as much to the point as any illustration can be. The 
thing omitted here from the discussion is clearly the 
one important thing. If we suppose that there is no 
inhabitant, we shall probably disbelieve the story, or 



SECT. 25.] LOGIC OF CHANCE. 309 

consider it to be exaggerated. If we suppose that there 
are inhabitants, the question is at once resolved into 
a &r higher one. The credibility of the witness is not 
the only element, but we should take into considera- 
tion the character of the supposed inhabitant, and the 
object of such an action on his part 

§ 25. I am aware that considerations of this 
character are often introduced into the discussion, but 
it appears to me that they are introduced to a veiy 
inadequate extent. It is oiften urged, after Paley, 
" Once believe in a Grod, and miracles are not incre- 
dible." Such an admission demands some extension. 
It should rather be stated thus. Believe in a God 
whose working may be traced throughout the whole 
moral and physical world. It amounts, in fact, to 
this; — ^Admit that there is a design which we can 
trace somehow or other in the course of things; admit 
that we are not wholly confined to tracing their con- 
nexion, or following out their effects, but that we 
can form some idea, feeble and imperfect though it be, 
of a scheme *. Paley's advice sounds too much like 
sajdng, Admit that there are &iries, and we can 
account for our cups being cracked. The admission is 
not to be made in so off hand a manner. To any one 
labouring under the difficulty we are speaking o^ this 
simple belief in a God almost out of relation to nature, 

* The stress which Butler lays upon this notion of a scheme 
is, I think, one great merit of his Analogy* 



310 LOGIC OP CHANCE, [CHAP. XIIL 

whom we then imagine to manifest himself in a 
perhaps irregular manner, is altogether impossible. 
The only form under which belief in the Deity can 
gain entrance into his mind is as the controlling 
Spirit of an infinite and orderly system. In fact it 
appears to me, that it might even be more easy for 
a person thoroughly imbued with the spirit of Inductive 
science, though an atheist, to believe in a miracle 
which formed a part of a vast dispensation, as the 
Christian miracles do, than for such a person, as a 
theist, to accept an isolated miracle. I repeat again, 
that I am not concerned at present with the origin 
of our belief in design or a Providential scheme. If 
any can find it in a study of physical science so much 
the better for them ; but whether it be obtained thence, 
or from moral and metaphysical grounds, or from Keve- 
lation, we must possess it, or miracles at the present 
day will be hard of credit. 

§ 26. It is therefore with great prudence that 
Hume, and others after him, have practically insisted 
on commencing with a discussion of the credibility of 
the single miracle, treating the question as though the 
Christian Kevelation could be adequately regarded as 
a succession of such events. As well might one con- 
sider the living body to be represented by the aggregate 
of the limbs which compose it. What I complain 
of in so many popular discussions on the subject is 
the entire absence of any recognition of the different 



SBcrr. 27.] logic of chance. 311 

ground on which the attackers and defenders of 
miracles are really standing. Proofs and illustrations 
are produced in endless variety^ which inyolving^ as 
they almost all do in the mind of one at least of the 
disputants, the very principle of causation the absence 
of which in the case in question they are intended to 
establish, they fail in the one essential point. To 
attempt to induce any one to disbelieve in the ex- 
istence of physical causation, in a given instance, by 
means of illustrations which to him seem only addi- 
tional examples of the principle in question, is like 
trying to stop the flow of a river by shovelling in 
snow. Such illustrations are plentiful in times of 
controversy, but being in reality only modified forms 
of that which they are applied to counteract, they 
change their shape at their first contact with the 
disbeliever's mind, and only help to swell the flood • 
which they were intended to check. 

§ 27. The bearing of the last few sections may be 
expressed as follows. Any one who believes that the 
moral and physical world form one great scheme need 
find no insuperable difficulty in accepting a Revelation 
which forms a portion of such a scheme, nor conse- 
quently in accepting the miracles, collectively and in- 
dividually, which are connected with the Hevelation. 
But i^ on the other hand, we start with the Inductive 
principle of uniform causation, and then attempt 
(leaving the notion of Providential supeiintendence 



312 LOOIO OF CHANCE. [CHAP. XJU. 

out of sight) to establish, first, such and such a miracle, 
and thence a Revelation ; it is hard to see how, on such 
principles, in the present state of feeling about scien* 
tific evidence, any accumulation of testimony could do 
more than baffle and perplex the judgment at the 
time, and leave us finally in doubt. 

§ 28. In this brief digression I have been able 
to do little more than state my own opinion. And the 
principal grounds on which it rests. Direct argument 
against those who should difier from me, I could scarcely, 
for the reasons already given, profess to offer. And 
here the matter must be left; for where men differ on 
such fundamental points speedy agreement is hopeless. 
A difference as to facts may be set at rest sometimea 
in a few minutes. But when the difference which 
separates party from party is one of ultimate principles^ 
approximation to one another may be indefinitely de^ 
layed. The suffrages of ages are required on matteni 
of this kind before a final judgment is obtained. 



In the remarks in § 1 1 it is implied that whatever may be 
the number of possible contingencies in any given case, the 
amount of onr belief in the assertion is the same, bdng thai 
fraction of oertainty (to use the common phrase) which is assign- 
ed by the figure which denotes the witness's veracity. If this be 
^, then, whether he say that a penny has given head, or that 
No. 79 has been drawn from a lottery of looo tickets, or make 
any other assertion whatever, we say that the chances are yV ^'■'^ 
the assertion is true. I am aware that the ordinary view ia 



raCT. 28.] LOGIC OF CHANCE. 313 

somewhat at Tariance with this, but the yariatioD is produced by 
what Beems to me a rather arbitrary assumption. It arises in 
the following way : One in ten of the assertions of the witness 
are false; but what will be the nature of these false assertions? 
I have assumed that we can tell nothing about their nature, so 
that when the witness does not tell the truth he may say any- 
thing. A common assumption, on the contrary, is that his false 
assertions must be confined in the above examples to telling 
numbers and throws, but telling these falsely. Hence, on my 
hypoihesiB, although some of the lies may undoubtedly take the 
form of asserting the number in question when it did not occur, 
yet these will, on the average, be quite inappreciable in nimiber, 
owing to the indefinite scope which the witness has for lying in 
other directions. On the other hypothesis, however, the number 
of these occasions on which an event is asserted without having 
happened will be very important. Out of 9990 occasions upon 
which any given number, say 65, is drawn, the witness will lie 
999 times. On my hypothesis no finite proportion of these lies 
will take the form of asserting 79 ; their effect is lost by their 
radiating out freely, as one may say, into space, and so being 
dissipated. If, on the other hand, we suppose them to be confined 
to the limits imposed by the nimibers in the lottery, their effect 
will be concentrated within these limits. On this view one 
out of these 999 particular lies will be a false assertion of the 
drawing of No. 79. And the same will apply when any of the 
other ntimbers are drawn. On this view therefore (imlike the 
one I have adopted) 79 will be announced when it did not occur 
in a certain regular proportion of cases in the long run. The 
assumption thus adopted seems to me arbitrary, but I do not 
think that the question is of any great practical importance. A 
brief discussion of some complications which are thus introduced 
will be found in Mill's Lo^c, Book lu. ch. 25. § 6. 



CHAPTER XIV. 
CAUSATION. 

§ 1. In several of the foregoing chapters we have 
been obliged to touch incidentally upon some of the 
philosophical and religious disputes into which the 
Science of Probability has at different times got itself 
embroiled. The interest and importance of these dis- 
putes however is so great that we must now enter into 
a more explicit and detailed examination of some of 
them. They almost all arise from the same source, — 
the bearing upon the doctrine of Universal causation 
of the assumption with which we started, and which 
was so fiilly discussed in our first two chapters. This 
assumption, the reader will remember, was that of a 
series as to the details of which we were in ignorance, 
whilst we possessed some knowledge about the average 
of the individuals of which the series was composed. 
The exact meaning of this assumption was« I hope, 
assigned with sufficient precision ; but as the doctrine 
of Universal Causation, which also enters into the 
conflict, has many different meanings, it will be neces- 
sary to determine accurately which of them is to be 
adopted here. 



SECT. 2.] LOGIC OF CHANCE. 315 

§ 2. I wish it to be understood then that we are 
about to enter into no metaphytiical or ontological dis- 
cussions; no enquiries will be made as to the intimate 
nature of causation, if it have any, nor need any of the 
associations excited by the term ^efficient cause' be 
introduced. We will employ the word simply in the 
sense which is becoming almost universally adopted 
by scientific men, viz. that of invariable unconditional 
sequence. 

It is in this sense that the word cavM is used by 
Mr MilL I refer to him in particular because his 
works contain one of the fullest and clearest explana- 
tions of the term with which we are now concerned. 
He points out indeed that by the ^antecedent' must 
be understood the 'sum total of antecedents/ explain- 
ing his meaning as follows; — ''It is seldom, if ever, be- 
tween a consequent and a single antecedent that this 
invariable sequence subsists. It is usually between a 
consequent and the sum of several antecedents; the 
concurrence of all of them being necessary to produce, 
that is, to be certain of being followed by, the conse- 
quent, lu such cases it is very common to single out 
one only of the antecedents under the denomination of 
cause, calling the others merely conditions." Several 
pages of explanation follow, devoted to tracing out 
the arbitrary and unphilosophical nature of any such 
distinction as that mentioned in the last clause, and to 
enforcing the &ct that throughout his work he shall 



316 LOGIC OF CHANGE. [CHAP. XIV. 

understand by cause ''the sum total of the conditions, 
positive and negative, taken together; the whole of 
the contingencies of every description which being 
realized the consequent invariably follows." (Mill's 
Logicj Bk. iil ch. v. § 3.) 

This meaning of the term is rapidly becoming the 
popular, or rather, the popular scientific one. There 
will of course be wide differences between different 
persons as to the extent over which they believe that 
causation in this sense can be predicated. All I ask 
is, that there may be no confusion; taking the above 
meaning of the word, let each assign its range of ap- 
plication according to his own opinion, but let us at 
least agree about its meaning. In logical language, 
having fixed the connotation of the word it must be 
left to the reader to assign the denotation. 

§ 3. Our first task will be to analyse the meaning 
of the term ; in doing so we shall find it, I think, 
neither so generally applicable nor so consistently 
applied as many persons seem to suppose. The two 
principal points to be discussed in the definition adopt- 
ed above seem to me to be the following ; — (1) The 
introduction of the 8ti/m4otaI of the antecedents into the 
cause; (2) the regarding the cause as the immedioUe 
antecedent. This latter condition is often not so ex- 
plicitly stated, but we shall easily see that it is impli- 
citly involved. 

§ 4. The first departure from the definition is in' 



SECT. 4.] LOGIC OP CHANCE. 317 

adopting the common practice of omitting some of the 
elements which conjointly form the invariable antece- 
dent. This omission is almost forced upon us when- 
ever we wish to make any use of the law of causa- 
tion; for as the cause was defined it appears to be 
barren and impracticable. It is defined in a hypo- 
thetical form, and the hypothesis is one that in most 
cases may not be, and in some certainly is not, realised. 
The cause is defined to be such a collection of antece- 
dents as t/* it were repeated the consequent would again 
foUow. But loill it be repeated 1 very seldom, perhaps 
never, if we insist on having all the antecedents accu- 
rately introduced. In very simple examples we do 
frequently find repetition, or something undistinguish- 
ably resembling it, but the moment we examine cases 
of any degree of complexity, though there may be 
repetition of many of the separate elements, the pre- 
cise combination is generally unique. This would be 
the case, for instance, with any thing which affected 
our bodies; for no man's constitution resembles exact- 
ly that of any one else. Still more would it be so 
if we were examining any of the greater operations 
of nature, such as thunderstorms, rain, Aurora Bo- 
realis. 

In these cases it would be true practically, and in 
almost all cases true theoretically, that the same 
antecedents never do actually recur. If so what be- 
comes of a definition which inyolves this hypothetical 



318 LOGIC OF CHANCE. [cHAP. XIY. 

repetition? It may stand as an expression of belief' 
of what would occur in contingent and possible cir- 
cumstances; but far more than this is requisite if any 
work is to be got out of the definition. It would be 
almost like defining a mortal as one who if he lived to 
be 150 would be in the last stage of decay. Such a 
definition as this is clearly barren and useless. In 
fact we seem to be reduced to the following dilemma ; — 
If we adhere rigidly to the sense of the term which 
was laid down at first no use can be made of it, for 
repetition then is certainly rare, and perhaps is not 
to be looked for at alL If we are to make any use of 
the formula we are forced to omit some of the ante- 
cedents, and then it ceases to be conformable to fact. 
It appears that we are driven to make our election 
between the useless and the Mae. 

§ 5. The reply would probably be that a suffi- 
ciently close approximation to real repetition to avoid 
all important error may frequently be found. This 
is certainly undeniable. It is obvious to all that 
events very similar to one another in most of their 
characteristics do constantly repeat themselves, and 
from this circumstance an abundance of inferences of 
great practical value may be obtained. But this is 
fidling short of the requirements of a Logic of Induc- 
tion. What we are at present concerned with is, not 
the looser form of the doctrine of Causation as it 
is practically made use of, but the strict form in which 



SECT. 6.] LOGIC OF CHAITCE. 319 

it appears as the basis of a science of inference about 
external things. 

§ 6. After a statement of the relation of cause 
and effect desciibed above, we are often reminded that 
there is another relation in which events may stand 
to one another, one dependent indeed on causation, 
but in which the antecedent, not being the immediate 
antecedent, cannot be considered the cause. The ante- 
cedent and consequent are, if one may so express it, 
not in contact in this case, but are a little removed 
from one another. The sequence is often a tolerably 
regular one, so as to present a certain degree of uni- 
formity, but not being an immediate sequence, it may 
generally be ultimately resolved into cases of causation 
by the discovery of intermediate links. These imi- 
formities are known by various names, such as, Em- 
pirical laws, or Uniformities dependent on causation. 
This distinction has been brought into notice in such 
recent disputes as those raised by the works of Mr 
Buckle and others of the same schooL When any 
uniformity has been observed in the conduct of men, we 
have been reminded that a mere generalization is a very 
different thing from the sequence of cause and effect. 
Mr Mill throughout his treatise seems to lay con- 
siderable stress upon this distinction; he differs indeed 
from Dr "Whewell in the language employed to de- 
scribe the distinction, but nevertheless speaks of it as 
<« one of the most fundamental distinctions in science; 



320 LOGIC OP CHAKCE. [CHAP. XIV. 

indeed it is on this alone that the possibility rests of 
framing a rigorous canon of Induction." 

§ 7. I cannot help thinking that this distinction, 
though having a foundation in nature, is, in the 
explicit form in which it is stated above, nothing but 
an example of the idolum farij arising from the neces- 
sities of common language, and deriving its principal 
support from common illustrations. Substitute for 
the time-honoured 'chain of causation,' so often intro- 
duced into discussions upon this subject, the phrase 
a 'rope of causation,* and see what a very different 
aspect the question will wear. For what is the con- 
ception necessarily conveyed by a chain 1 that of 
determinate distinct links following one another in 
succession, of stages marked off from one another in 
nature as well as distinguished in language. Where- 
as what we really find in nature is an evolution 
rather than a succession; the stages when examined 
at a little distance from one another are tolerably 
distinct, but, when closely examined, each merges into 
the next and blends with it by insensible degrees. 
They are like the strands of a rope; at a little distance 
there may be what one might call successive patches 
or stages, but when we look closer we find that each 
sti*and continues without the slightest break of con- 
tinuity. Instead of having links definitely marked out 
for us, the steps and stages have to be assigned by our- 
selves, and have a great deal of what is arbitrary in them. 



SBGT. 8.] LOGIC OF CHANCE. 321 

§ 8. This mistake finds much oountenance in 
the oommon practice of using letters of the alphabet 
to denote the causes and effects. The following is a 
fietir sample of this mode of illustration. < Within the 
limits of past experience A has been always followed 
hy Bf A therefore is probably the cause of B; still 
there may be some intermediate link (7, between A 
and B, which is respectively effect of A and cause of 
B. Or even eventually a D may be discovered be- 
tween G and B, and so on.' I do not, of course, give 
the above as a sample of the reasoning employed, but 
simply of the phraseology in use. It involves through- 
out the conception of successive links of a chain, the 
only doubt ever felt being that a step or two may have 
been overlooked, though, if so, these steps also will 
be links. But is this conception consistent with £Bict1 
Sorely it needs but a very little reflection to perceive 
that all these stages, which we thus mark out, exist 
as distinct stages only in our classification, and that 
when we look at nature we find, not an A and B as 
successive links with a possible intruder G between 
them, but rather an A and B as successive portions 
of a strand between which, if we had chosen, we 
might have interpolated an indefinite number more. 
A dose of arsenic will cause death, but how many 
intermediate stages are there between these? Even 
where the cause and effect seem most proximate, as 
for instance in an explosion giving rise to the sensa- 

21 



322 LOGIC OF CHANCE. [CHAP. XIV. 

tion of sound, W9 might if we pleased interpolate any 
quantity more of what are called links. 

§ 9. All this is too obvious to have escaped 
notice, but its bearing on the distinction between 
laws of causation and empirical laws has been 
generally neglected. For if the above remarks be 
true, this distinction vanishes; it vanishes, at least, 
as an accurate theoretical distinction, though it may 
be retained in a looser form for practical purposes. 
It is essential to the existence of a cause, as an un* 
conditional invariable antecedent, that it should imme- 
diately precede the effect; if there is any interval we 
can never insure the succession from being frustrated 
by the intrusion of some counteracting agency. I am 
aware that such counteracting agency is often ex- 
pressly excluded, but I cannot help regarding this ex- 
clusion as being entirely unwarranted. By what right 
is the exclusion maintained, when the object is to 
prop up and insure from failure a law of causation, 
but refused when wanted to perform the same service 
for an empirical law? Only exclude counteracting 
agencies and any observed empirical uniformity will 
be as * unconditional' and invariable as can be desired; 
suffer these agencies to enter and the laws of causa- 
tion will cease to possess these characteristics. If we 
admit any interval between the antecedent and con- 
sequent, and determine to be consistent and equitablo 
towards the two different kinds of uniformity in 



SECT. 10.] LOGIC OF CHAKCE, 323 

question, we shall find that no theoretical distinction 
«an be maintained between them. A man takes arsenic 
and dies; that was the 'cause' of his death, for the 
sequence is an invariable one, if no counteracting 
forces are at work. I kill a horse in a hot climate 
and it is eaten up by flies; this is only a 'uniformity,' 
but why? keep out the counteracting forces, and this 
sequence also will be invariable. Of course if any 
one beats off the flies the horse will not be devoured, 
but so if a surgeon comes with a stomach-pump the 
man may recover. We quite admit that in l^e one case 
the stages are many, and any of them can be interfered 
with; this is abundantly sufficient to establish an im- 
portant practical distinction ; but inasmuch as in the 
other case also interference is possible, no theoretical 
distinction can be based on this. If we shift onr 
ground and claim the right to introduce a (7, by declar- 
ing that the real cause and effect were not the arsenic 
and the death, but the arsenic and some intermediate 
change, we shall not secure the position ; there would be 
the same necessity as before for excluding counteraction. 
§ 10. The conclusion from the above investigation 
seems to be that unless we determine rigorously to 
adhere to an immediate antecedent we can never secure 
an unconditional and invariable one. It will not take 
much proof to show that such a determination would at 
once make the formula of universal causation barren 

and impracticable. 

21—2 



324 LOGIC OF OHANOS. [CHAP. XIY. 

For what is an immediate antecedent 1 not the 
next link in the chain, for of links none are to be 
found in nature, but the next point in the strand of 
rope; and what is this) Take two points as near to 
one another as we please, another can always be in- 
terposed between them; examine any two stages in 
the sequence of phenomena, and any number more of 
intermediate stages can be conceived. The notion of 
an immediate antecedent can give us nothing, when 
strictly examined, but the tendency and magnitude of 
the forces in action at the point of time in question, 
but not the condition of things at any previous finite 
interval; it tells us only the form or law of develop- 
ment then and there, it does not give us successive 
stages of that development To borrow a mathe- 
matical illustration, we can only determine, by means 
of this notion of immediate succession, the direction 
of a curved line at a given point, but we cannot 
discover any other point on the line however near 
to the given point. 

§ 11. The notion then of invariable antecedence 
and consequence, when the antecedent and consequent 
are really immediate, seems to dwindle down to 
this; — Given the state of the phenomena at any given 
time, it declares the only phase of development which 
those phenomena can assume at that time, but it does 
not enable us to infer certainly what will be the 
condition of those phenomena after any finite interval. 



SECT. 12.] LOGIC OF CHANCE. 325 

however brief we may suppose that interval to be. 
k Hence the barrenness of the formula; for telling us 
only what things are becoming, and not what they 
have been or will become, no real inferences about the 
future can be made by means of it. If B were the 
next link on a chain we could infer its presence fix>m 
that of A by the law of causation, but if J3 be only an 
immediately neighbouring position on a sti'and that 
law will simply give us the tendency of things &t A; 
we may thus give a tolerable guess as to whether B 
will follow or not, but we cannot certainly infer the 
&ct. 

§ 12. The general bearing of the last few sec- 
tions upon the doctrine of causation seems to lead us 
to the following conclusion. In investigating nature, 
80 far as we can without prepossessions, we come 
upon a large number of different successions which 
in their main features resemble one another. If we 
insist on introducing all the antecedents of any suc- 
cession we must admit that the succession will be in 
almost every case unique, but there iis a degree of de- 
termination short of this which enables us without 
appreciable error to speak of the same succession 
recurring repeatedly. When we do so speak, however, 
we find that the formula of Causation is still open 
to further and more serious objections. Each suc- 
cession is composed of more or less of what are rough- 
ly called links of a chain. The more nearly in inti- 



326 LOGIC OP CHANCE. [cHAP. XIV. 

mate succession we suppose these links to be, the more 
nearly inyariable does the succession tend as a general 
rule to become. This tendency thus obsenred over a 
certain extent, is greatly enlarged by Analogy, until 
ire describe the immediate antecedent and consequent 
83 forming an invariable unconditional succession* 
Of the doctrine, however, in this form, no use 
whatever can be made. For all practical purposes 
the cause must be understood in a sense in which 
the succession not being accurately immediate, is not 
really unconditional. 

§ 13. So much then for the meaning of the term 
Causation. We seem led to the conclusion that it is 
an ambiguous term, ha ving two senses , one an ideally 
precise, but almost useless sense, the other a rough 
sense adapted foFworking purposes We shall now 
be better able to examine into the nature of the 
conflict, or supposed conflict, between this law, in 
either of the above senses, when it is generalized into 
a universal formula, and the assumptions which under- 
lie the science of Probability. 

§ 14. The following is the principal form which 
this conflict assumes. It has ali^ays been felt with 
more or less clearness that ignorance of the details, as 
combined with knowledge of the averages, is insepara* 
bly connected with the notion of probability. Hence 
arises anxiety in studying probability, lest the admis- 
sion of this ignorance should be supposed to carry 



SBeiT. 15.] LOGIC OP CHANCE. 327 

along with it, as the ground of our ignorance, the as- 
sumption that the individual events in question can 
happen without causes. In most works upon the 
subject, therefore, whenever a discussion aidses about 
our ignorance of particular events, whenever in fact 
the word chance has to be introduced, it is generally 
considered necessary to utcer a caution against our 
believing in there really being such a thing as chance* 
Hume, for instance, in his short Essay on Probability, 
comqiences with the remark, ''though there be no such 
thing as chance in the world, our ignorance of the 
real cause of any event has the same influence on the 
understanding 4cc.'* Such a caution as this has been 
especially insisted on by those who have written ex- 
press treatises on Probability. I hardly know of one 
in which there is not inserted, somewhere at the out- 
set, an emphatic disavowal of any belief in chance. 
Professor De Morgan goes so far as to declare that the 
foundations of the theory of Probability have ceased 
to exist in the mind that has formed the conception 
" that anything ever did happen or will happen with- j 
out some particular reason why it should have been j 
precisely what it was and not anything else," Some- ' 
what similar remarks might be quoted from Laplace 
and others. 

§ 15. The view above described refers principally to 
the natui'al and physical sciences. It there occupies 
rather a defensive position, the fact being insisted on 



328 LOGIC OF CHANCE. [CHAP. ZIV. 

that vhenever in these subjects we maj happen to 
be ignorant of the details we have no warrant for 
assuming in consequence that the details are un- 
caused. But the corresponding yiew takes up a much 
more aggressive position when applied to social sub- 
jects. Here the attempt is often made to prove 
causation in the details from the known and admitted 
regularity in the averages. A considerable amount of 
controversy has been excited of recent years upon this 
topic, in great part owing to the vigorous and out- 
spoken support of the Necessitarian side by Mr Buckle 
in his History of GivilizctUon, 

§ 16. It should be remarked that an attempt is 
sometimes made in these cases almost to startle the 
reader into acquiescence by the singularity of the ex- 
amples chosen. Instances are selected which, though 
they possess no greater logical value, are, if one might 
so express it, emotionally more powerful That the an- 
nual number of suicides should remain nearly the same 
is assumed to be strange enough, but what are we to 
say to the staggering fact that the number of mis- 
directed letters annually sent to the post-office is about 
the same? Laplace himself scarcely dares to say more 
than that "he has heai*d that this is the case;" and 
writers of such repute as Dugald Stewart seem to 
have found real satisfaction in the fact that his asser- 
tion is after all only a hearsay. 

§ 17. The aim of all such attempts is the same. It 



f 



SECT. 17.] LOaiC OF CHANCE. 329 

I is hj the help of statistical uniformity to establish the 
existence of causation (in the sense of invariable 
unconditional sequence) in indiyidnal cases. I must 
confess, in spite of Professor De Morgan's assertion, 
that I cannot see that the matter, whicheyer way it be 
settled, has necessarily much to do with Probability. 
The caution no doubt, in the connection in which it 
generally occnrs, may be a very useful one, for the 
opinions of the vulgar about the occurrence of events 
in games of chance is utterly vague and unscientific. 
But as a contribution to the theory of the subject I 
cannot help r^arding it as needless, and even calculated 
to mislead. Our reason for employing the theory of 
Probability is our ignorance of the single events ; but 
I cannot see that it is of the slightest importance from 
what cause this ignorance arises. It may be that 
ignorance is unavoidable from the nature of the case^ 
there being no regular connection between antecedent 
and consequent ; the causative link, as one may say, 
having been snapped. It may be that such a 
connection is known to exist, but that either we 
cannot discover it or that its discovery would be 
troublesome. It is the fact of this ignorance that 
makes us resort to the theory of Probability, the 
causes of it are quite irrelevant When we do not 
know the events, considered singly, and choose our 
method just because we do not, it seems to me a mere 
digression to insist upon the &ot that there is no 



330 jioaic OP CHANCE. [chap. xnr. 

essential hindrance to our knowing them, and that we 
inight do so were our faculties sharper than they are. 

I am quite aware that on the view of Probability 
adopted by Professor De Morgan, the question assumes 
a somewhat different aspect. He, in common with 
many writers on the subject, seems to claim that the 
amount of our belief about the single event must 
admit of justification. My reasons for dissenting 
from this view have been already fully given; I need 
only therefore remark that if the view adopted in this 
Essay be correct we are absolved from any such justifi- 
cation, and are therefore perfectly indifferent as to 
what view may be taken about the single event. We 
are concerned only vith averages, or with the single -r* 
event as deduced from an average and conceived to 
form one of a series. We start with the assumption, 
grounded on experience, that there is uniformity in 
this average, and, so long as this is secured to us, we 
can afford to be perfectly indifferent to the fate, as 
regards causation, of the individuals which compose 
the average. 

§ 18. When thus viewed the question to be 
decided assumes a rather different form. It can only 
be stated thus, Is the assumption mentioned in the last 
paragraph an impossible one under the carcumstances? 
Or, by assuming that events of any kind display a 
uniformity in the long run, are we precluded from 
admitting that any or all of these events had no 



SECT. 19.] LOGIC OF CHANCE. 331 

regular antecedents or conaeqnente? Let us take aix 
^i^ample. We know from experience that when a 
penny is tossed up a great many times, heads and tails 
occur in about equal numbers. On the view now 
under discussion it is maintained to be quite essential 
that the result of each separate toss should have its 
invariable antecedent and consequent. I do not deny 
that this, as a distinct fact, may be true, but simply 
that it has any necessary connection with the previous 
assumption. For let us suppose that some or all of 
the throws had no invariable antecedents; what then I 
The fact of the general regularity being undeniable, 
the objector would have to assert that such a suppo- 
sition was an impossible, or rather an inconsistent one. 
What is demanded is the proof by which he shows it 
to be inconsistent. This is surely no unreasonable, 
demand, especially when we bear in mind the fact that 
the two doctrines, thus supposed to be inconsistent^ 
have as a matter of fact constantly existed together in 
apparent harmony in the same minds at the same 
time. The harmony may be illogical, but if so, this 
should be distinctly proved. Millions, for instance, 
have believed in the uniformity of the seasons, who 
certainly did not believe in, and perhaps distinctly 
denied, the existence of necessary sequences in all the 
phenomena of each particular season. 

§ 19. If we recur to the enquiry entered into in 
the earlier part of this chapter we shall find, I thinkj,^ 



332 LOGIC OF CHAKCE. [cHAP. XIT. 

that the conoeption of causation generally employed 
in these discussions fails in both of the respects to 
which attention was there drawn. As regards the 
introduction of aU the antecedents into the cause, this 
is necessarily the case. Were this introduction in- 
sisted on, the sequence, as I have already remarked, 
would often be almost unique. Seldom or never 
should we be able to obtain enough instances to form 
statistics, except by neglecting a very considerable 
portion of the combination of antecedents. Statistics, 
from their nature, preclude any but a very slight 
degree of specialization. So also as regards the im- 
mediateneaa of the connection between cause and 
effect. So far from this being secured, the connexion 
in most statistical tables is of the loosest possible de- 
scription. We there have set before us ^'s and B's 
with a very appreciable amount of separation between 
them. Nothing could be inferred in this way that 
would really bear upon so intimate a connection as 
that between the final form of the antecedent and the 
initial form of the consequent. The elements here 
under consideration are altogether disparate; we might 
as well attempt by reasoning about the separate links 
of a chain to draw conclusions about the molecular 
constitution of the iron, or discover whether the links 
were in actual physical contact or not. 

It may be remarked that we are speaking here of 
what statistics might be conceived to be capable of 



SECT. 20.] LOGIC OF OHANCB. 333 

provifig, not of what could be inferred from such 
tables as actually exist. The^e for the most part do 
not even offer anything that can be considered to be 
a sequence of ^'s and ^'s. Facts connected with 
one element only are laid before us, it may be thefts 
or murders or suicides. These actions, no doubt, may 
have their causes and their effects, but before a 
sequence of any kiud, whether complete or incom- 
plete, near or remote^ can be inferred from such 
statistical tables, we must have other tables before us 
which shall refer to the supposed regular antecedents 
and consequents alike. 

§ 20. It may £urly therefore be asked here 
whether the opinion that statistics have added to 
the extent over which causation can be shown to exist, 
is entirely incorrect. Briefly stated, my own view 
upon this subject is as follows; — It is only by means 
of causation that we are able as a general rule to 
make individual inferences about natural phenomena 
of any kind; that is, if we want to know what will 
happen under any circumstances we can only do this 
by ascertaining what has happened under the same or 
nearly similar circumstances before, and then nniLlring 
the assumption that the antecedents being the same 
the consequents also will be the sama In other words, 
inference is attainable, either actually or conceivably, 
wherever causation prevails, and not elsewhere. 

Now what follows, as a natural consequence of 



334 LOGIC O^ GHAKCE. [CHAP. XlV. 

this, upon the discovery of statistical regularity ? Take 
the instance of suicides. As regards the individual 
crime no certain inference whatever is possible. If 
the man's actions have their regular sequence of cansd 
and effect most of the elements which combine to mtike 
any one antecedent are unknown to us. Hence the 
impression would not unnaturally arise that inferences 
would be equally unattainable in the case of the 
average. To give a numeiical illustration, many per* 
sons would suppose that the numbers of suicides in 
successive years in London would present a series of 
about as great irregularity as the following; — 1, 200, 
50, 700, 3, 150 &c. For it is only through such irregu- 
larity that prediction would be precluded when we had 
a seiies of single events of this kind in contemplation. 
What we really find however is that the number re^ 
mains about the same year by year. Hence inferences 
can be drawn, not indeed about individuals but about 
averages. And since in the former case, when the 
reference was to the individual instance, inference 
implied causation, it is assumed that the same must 
hold true in the latter case also. 

§ 21. Of course if the meaning of causation be 
extended so as to include any kind of regularity 
whatever that enables us to make inferences, its ex- 
istence can doubtless be proved by means of statistics; 
but this seems to be using the term in a sense very 
different from that of invariable unconditional sequence 



SECT. 21.] LOGIC or CHANCEr 335 

with which we started. The popular ideas upon the 
subject would perhaps be expressed bj saying that 
the regularity which is detected proves the events 
not to happen at random but to be under some kind 
of control. Turn such phrases as we may I cannot 
see more in them than a restatement of the fact that 
the observed regularity does exist; imless indeed the 
distinct error be involved that the regularity must 
somehow be produced by a kind of compulsion, so 
that causation in physical matters involves regularity 
produced by restraint. This would of course be to 
wander far from the scientific meaning of the term. 

When we have observed the regularity in question 
for some time we are undoubtedly disposed to extend it 
fiirther. If we find, for example, that a certain number 
of thefbs have been committed each year for some time, 
we expect that the same state of things will continue 
for some time to come. In other words, we assume a 
certain degree of order or stability, in the operations 
of nature, and the statistics^ if continued, confirm this 
assumption. As I have already said (Ch. vii. § 7), 
causation in this sense is certainly necessary for our 
inferences, and statistics continually prove its exist- 
ence. But no amount of regularity of this kind seems 
to me to bring us nearer to proving that each separate 
event comprised in the statistics has its invariable and 
unconditional antecedents, which is the point at pre- 
sent in question. 



336 LOGIC OF CHAHGB. [CHAP. XIT. 

§ 22. The proof of this latter fact should surelj 
be sought, not in the regnlarily of the statistics, bat 
in one sense in their irr^ohiTitj. By finding that 
the number of thefts or suicides remains nearly the 
same we learn little or nothing about the question 
at issue; but if we found that every alteration in their 
number was connected with some alteration in such 
antecedent or concurrent circumstances, as the vigilance 
of the police, the happiness of the poor, or their politi- 
cal, moral, and religious progress, we should then b^in 
to entertain a strong belief that some or all of these cir- 
cumstances were causes of the observed and recorded 
phenomena. I only remark this in passing, to pursue 
it further would be to wander from our proper province. 

§ 23. The mere regularity of the observed statis- 
tics, on the other hand, seems to me scarcely to have 
any connexion with causation, in the strict sense in 
which we have defined the term above, but to lead to 
an entirely distinct class of conclusions. It is in this 
way that the £act is ascertained, which was described 
and illustrated in the eleventh chapter, that almost all 
the properties of natural classes of objects preserve a 
general uniformity amidst individual variations. Thus, 
for example, although we cannot tell beforehand what 
may be the height, weight, &o. of any given man, we 
know that by continuing our observations over a suffi- 
ciently large extent we shall find these elements not 
only preserving tolerably nearly a certain average, 



1 



SECT. 24.] LOGIC OF CHANCE. 837 

but grouping themselves in a regular and orderly 
way about this average. It is a very early result of 
observation to detect this r^ularity in the simplest 
properties and qualities of natural classes ; what sta- 
tistical uniformity seems to me to establish is, that 
a similar uniformity prevails in all the most recondite 
of these properties, and in almost all their remotest 
consequences. Take but one instance, — ^the observed 
&ct that the number of misdirected letters remains 
about the same year by year. This appears to me 
to establish exactly the same kind of conclusions as 
the observed fact that the lengths of the lives of large 
bodies of men remain about the same. Just as obser* 
vations of the latter kind show that the strength of 
constitution in different men preserves a tolerable 
regularity, so do those of the former kind show that 
there is a tolerable regularity in the strength of their 
memory. In each case the restdt is complicated by 
the cooperation of many external agencies to produce 
the observed result; statistics show also that these 
agencies themselves present a similar regularity. 

§ 24. That which gives to discussions of this 
kind their chief interest is, no doubt, their supposed 
bearing upon the vexed question of the freedom of 
the will. I have no intention of entering further into 
this question than is necessarily involved by opposi- 
tion to a line of proof which is often adopted in the 
discussion. I am, in ^uct, simply opposing an argu- 

22 



338 UMJIG OF CHAHCB. [cHAP. XIY. 

ment which is lued against a particular doctrine; 
npon the independent tmth of the doctrine itself I 
have no intention of expressing an opinion here. The 
connection between antecedent and conseqnrait — in 
this case the determination of the will — ^wonM pro- 
bably be nniyersallj admitted to be of the most 
intimate kind; the error of what for want of a better 
name must be called the school of Mr Badde is» it 
appears to me, to attempt to prove a connection of 
this kind from statistics^ which at best are only con- 
cerned with the less intimate connection of the eventSy 
that is, with the looser sense of the term. This refers, 
of course, to such inferences as may be drawn from 
the mere uniformity of the numbers of persons who 
perform certain acts^ not (as already remarked) to a 
critical examination of the deviations from this uni- 
formity. 

§ 25. Before completing this chapter, I must 
make a brief reference to certain phases of some of 
the forgoing objections, which are theological rather 
than philosophical They are howeyer of no great 
present interest^ since they are scarcely likely to be 
urged now in any form in which we can take notice 
of them here. 

They are supposed inferences from Probability in 
favour of Atheism, and are of two kinds, which are 
distinct and indeed almost contradictoiy of one an- 
other. We will examine them ia turn. 



\ 



SECT. 25.] LOGIC OF CHA17CE. 339 

(1) An objection to Probability, which was once 
popular and which I suppose still lingers in some 
quarters, is that it refers events to chance. If we spelt 
this word with a capital 0, and implied that it was re- 
presentative of a distinct creative or administrative 
agency, we should, I presume, be guilty of some form of 
Manichseism. The only other meaning of the objection 
can be, that we assume that events (some events, that 
is) happen without a cause, and therefore remove 
them from the control of the Deity. But, as already 
pointed out, this is entirely a mistake. The science 
of Probability makes no assumption whatever about 
the way in which events are brought to pass, whether 
by causation or without it. It is simply a body of 
rules applicable to classes of cases in which we do 
not or cannot make inferences about the individuals ; 
it commits itself to no opinion as to the ground of our 
inability to make such inferences. The objection there- 
fore appears to amount to nothing more than this, 
that the assumptions upon which the science of Pro- 
bability rests, and in consequence of which it is 
employed, are not inconsistent with a disbelief in 
causation within certain limits, causation of course being 
-^^ understood simply in the sense of regular sequence. So 
expressed the objection would (on my view) be perfectly 
true, though what bearing it could have upon Atheism 
is not easy to be seen, and must be left to any who urge 
it to explain. 

22—2 



1 



340 LOGIC OF GHAirCB. [CHAP. XIV. 

§ 26. (2) The other objection is almost exactlj 
the reverse of this, but is so utterly unreasonable that 
there is some difficulty in believing that the writezB 
who have urged it really meant what they said. As 
it has been already alluded to in an extract from 
Laplace*, I will refer agaip to the example which ha 
has there given. A man is supposed to have observed 
the fact that male and female births occur in the 
long run in a nearly constant ratio, and he has 
attributed this to Providence. All that need be meant 
by this is, that afber he had observed the ^t he came 
to the conclusion that the Creator had so arranged 
our frames that the births should occur on the average 
in this ratio. He then studies Probability, and ascer- 
tains that he had been labouring under a mistake, 
for that these proportions and numbers were in reality 
nothing but the 'development of the respective pro- 
babilities.' As I have stated before, if by the pro- 
babilities be meant the proportions, the assertion is 
tautological If it be meant that there is something in 
our constitution by which the particular births are so 
brought about that oq the average they occur equally 
often, the reply would of course be that this constitution, 
like everything else in our fiume, had been produced 
mediately or immediately by our Creator. The phy- 
siologist believed already that there was some arrange- 
ment of our bodies by which each individual birth 
* Chap. n. § 8, 



SBOT. 27.] LOGIC OF CHANCE. 341 

was determined; unless the statistical data lead him 
hj some means to detect this law, he knows absolutely 
nothing hj the numbers that he did not know before. 

The fitct is that Probability has little more to do 
with Natural Theology (either for or against it) than 
the principles of Logic or Induction have. It is I 
simply a body of rules for drawing inferences about 
classes of events which are distinguished by a certain { 
quality. The believer in a Deity will, by the study 
of nature, be led to form an opinion about His works, 
and so to a certain extent about His attributes. But 
to propose that he should abandon this belief because 
the sequence of events, — not, observe, their tendency 
towards happiness or misery, good or evil, — is brought 
about in a way different from what he had expected, is 
so extraordinary a statement, that there is a difficulty 
in supposing that one has fully understood it. 

§ 27. It is both amusing and instructive to con- 
sider what different feelings might have been pro- 
duced in our minds by this connexion between, what 
may be called, individual ignorance and aggregate 
knowledge, had they presented themselves to our 
experience in a reverse order. Being utterly unable 
to make predictions about a single life, or the conduct 
of a single person, men are startled and sometimes 
terrified at the discovery that such predictions can be 
made when we are speaking of large numbers. And 
so some exclaim, This is denying Providence I It is 



342 LOGIC OP CHANCE. [CHAP. XIV. 

utter Fatalism ! Now let us assume, for a moment, 
that our first acquaintance with the subject had been 
with the aggregate instead of the individual lives. 
We might conceive of something approaching to this 
in the case of an ignorant clerk in a Life Insurance 
Office, who had never thought of life except as having 
such a ^ value' at such an age, and who had hardly 
dealt with it but in the form of averages. Can 
we not conceive his being astonished and dismayed 
when he first realized the utter uncertainty in which 
the single life is involved? And might not his excla- 
mation be, in tum^ Why this is denying Providence ! 
It is utter Chaos and Chance t A belief in a Creator 
and Administrator of the world is not confined to 
any particular assumption about the sequence of 
events, but those who have been accustomed to regard 
events imder one of the aspects above referred to, 
will often for a time feel at a loss how to connect 
them with the other. 



CHAPTER XV. 

ON STATISTICS AS APPLIED TO HUMAN 
ACTIONS. 

§ 1. Throughout this Essay examples have been 
drawn almost indifferently both from purely physical 
phenomena and from those which are concerned di- 
rectly with human actions; in the case of the latter^ 
moreover, some of our examples refer to conduct which 
is purely voluntary, whilst in others the human will 
is but a remote cause of the effects described. It is 
now time to enter into a short enquiry as to how far 
it is right thus to put these voluntary actions upon 
the same footing as the results of the seasons or the 
turning up of the &ces of a die, and to subject them 
all alike to the same rules. 

The enquiry before us is, of course, but a limited 
portion of a far wider enquiry which has been much 
debated of late years, namely, whether what we have 
termed Phenomenalist or Material Logic is as applica- 
ble to the facts of society as it is generally admitted to 
be to those of inanimate nature. It is needless to say 
that nothing professing to be an adequate investigation 
of such a subject as this will be made in the present 
chapter; but the enquiry is one which must have been 



544 LOGIC OF CHANCE. [CHAP. XV. 

SO ofben suggested in some of the previous cliapters, 
that it cannot be altogether passed over hera In 
Probability we are concerned only with a limited 
portion of human conduct^ with actions, that is, which 
show some uniformity when arranged in a series, and 
with the inferences which can be drawn concerning 
them; but inasmuch as the criticisms which will 
presently be offered apply equally to drawing inferences 
about human actions of almost any kind, it will be 
simpler to commence our enquiry under the latter or 
more general form. 

§ 2. It has been already repeatedly stated that 
the standing point occupied by the observer who ia 
supposed to make the inferences we have been consider- 
ing, is that in which he looks out on to things which 
are happening about him. He is supposed to observe 
coexistences and sequences of things around him, 
which he then proceeds to classify, and from which he 
draws what inferences he can. To retain such a 
standing point consistently two conditions, amongst 
others, seem to be presupposed. These are (1) That 
the observer should leave the things which he observed 
to work out their courses undisturbed by any inter* 
fi^ence on his own part. (2) That he should adhere 
consistently to the position of an observer, and not 
in imagination step down and take a place amongst 
the things which he observes. In the attempt to 
construct the Logic of Society, or Sociology as it is 



SECT. 3.] LOGIC OF CHAKCK. 345 

oflen termed, both of the above conditions seem to me 
to be often neglected. The neglect of the former ia, I 
think, an inherent imperfection in any snch science of 
human conduct; that of the latter is rather a fiJlaey 
into which loose thinkers are apt to faD. We will 
examine these conditions in turn. 

§ 3. (I.) * To say that the objects of any kind whose 
behaviour we are considering are to be left free from 
any interference on our own part, is to make a claim 
which is so obviously demanded, that the caution may 
seem unnecf^ssary. And it certainly is not needed in 
the case of most inferences about inanimate objects. 
Any per8<>n can see that to draw inferences about a 
thing, and then to introduce a disturbance which was 
not contdmplated when the inference was drawn, is to 
invalida-;^ the results we have obtained. But when 
the inference is about the conduct of human beings it 
is ofter^ forgotten that in the inference itself, if publish* 
ed, we may have produced an unsuspected source of 
distuj.'banca In other words, if the results of our 
invejtigations be given in the form of statements as to 
what people are doing and what they will do, the 
Fioment these statements come before their notice the 
a^nts will be subject to a new motive which will 
produce a disturbance in the conduct which had been 
inferred. We may make what statements and criticisms 

* The six sections which follow contain the substance of an 
article published by me in Frater'i Magazine for May i86i. 



346 IX)OIC OF CHANCE. [c&AP. XV. 

we please about ihepast conduct of men, but direcUy 
we commit ourselves to any statements about the fu- 
tuie, or, in other words, begin to make predictions, we 
lay ourselves open to the difficulty just mentioned. 
That predictions can be made seems to be held by most 
of those who have adopted the application of logic now 
under consideration. They do not, of course, claim to 
be able to foretell the particular actions of individuals, 
but they constantly assert that it is quite possible that 
we may some day be able to foretell general tendencies, 
and the results of the conduct of large masses of men. 

§ 4. The following extracts from Mr Mill's Logic^ 
Bk. VI. ch. iiL § 2, will contain the best description of 
these claims of Sociology. After referring to the con- 
dition in which astronomy once was, and the science 
of the tides now is, he describes in the following wo/ds 
the practical aims of Sociology and the ideal perfection 
of the science, from which we are precluded only by 
the imperfection of our Acuities: — ''The science of 
human nature is of this description. It &lls &r short 
of the standard of exactness now realized in Astro- 
nomy; but there is no reason that it should not be 
as much a science as Tidology is, or as Astronomy 
was when its calculations had only mastered the main 
phenomena, but not the perturbations. 

"The phenomena with which this science is conver* 
sant being the thoughts, feelings, and actions of human 
beings, it would have attained the ideal perfection of 



SaCT. 6.] LOGIC OF CHAKCE. 347 

a science if it enabled ns to foretell how an individual 
would think, feel, or act^ throughout life, with the 
same certainiy with which astronomy enables us to 
predict the places and the occultations of the heavenly 
bodiea" 

§ 5. It will hardly be denied that there is the 
following distinct theoretical objection to the abov« 
illustration. The publication of the Nautical Alma- 
nack is not supposed to have the slightest effecfc upon 
the path of the planets, the publication of any pre- 
diction about the conduct of human beings (unless 
it were kept out of their sight, or expressed in un- 
intelligible language) almost certainly would have some 
effect. The existence of this distinction renders all 
physical illustrations of any kind whatever entirely 
inapplicable when we thus attempt to explain the way 
in which it is supposed that human conduct can be 
studied and foretold. 

§ 6. I wish it to be clearly understood that we 
are not here getting involved in any Fate and Free- 
will controversy; the objection before us does not arise 
out of the foreJcTundedge, but out of the foreteUmgj 
of what the agents are going to do. Assuming that 
the abstract possibility of foreseeing human conduct, 
alluded to in the extract above quoted, is quite com- 
patible with our practical consciousness of freedom, 
I maintain that a difficulty of an entirely distinct 
character is introduced the moment we suppose that 



348 LOGIC OF CHAircB. [chap. xv. 

this conduct is foretold, or rather, if one may use 
the term, forepMUhed, After all the causes have 
beeu estimated which can affect the agent, with the 
single exception of the sociological publication which 
describes his conduct, we shall perhaps find that the 
result is subsequently falsified by the disturbing agency 
of this publication itself. 

This disturbance, observe, is not of the nature of a 
mere complication of the result; it takes the form of a 
distinct contradiction^ Something was going to be 
done, and was therefore announced ; in consequence of 
the announcement that thing is not done, but something 
else is done instead* But had this further consequence 
been foreseen (as we must, on our present assumption, 
suppose might haye been the case) and allowed for, we 
still shall not find any escape from the difficulty. Were 
this all we had to take into account we should have 
nothing further to apprehend than a complication; but 
beyond all this there is the conflict between the final 
announcement and the conduct announced, which can- 
not be avoided I repeat again, that it is not fore- 
knowledge, but foretelling, that creates the difficulty ; 
the observer, after he has made lus announcement^ or 
whilst he is making it, may be perfectly aware of the 
effect it will produce, and may even privately com- 
municate the result to others, but once let him make it 
so public that it readies the ears of those to whom it 
refers, and his work is undona His position, in £em^ is 



SSCT. 7.] WQIO OF CHAKCB. 349 

somewhat like that of Jonah at Nineyeh, OiTing him 
the fullest reoognitioix of his power of foreseeing things 
as they would actuallj happen, we must yet admit that 
lie labours under an inherent incapacity of publicly 
announcing them in that form. The city was going 
to be destroyed; Jonah announces this; in consequence 
the people repent and are spared. But had he foretold 
their repentance and escape, the repentance might 
never have taken place. He might, of course, make 
a hypothetical statement, so as to provide for either 
alternative, but a categorical statement is always in 
danger of causing its own apparent falsification. 

§ 7. The only reply, I think, can be that al- 
though the above difficulty is a theoretic objection, 
the effects referred to will be so utterly insignificant 
that they can be neglected in practice. But it is surely 
very doubtful whether distinct statements about 
human beings can be expected to produce little or no 
effect upon their conduct The magnitude of this dis- 
turbing effect would seem to depend in gi*eat part 
upon the particular aimouncement made, and the 
intelligence of the agents referred to in it. If the 
announcement is concerned with matters of little im- 
portance, or with the conduct of persons who for any 
reason are not likely to take notice of what is said 
about them, then the considerations to which I have 
been referring might be n^lected without serious 
eitbr. We might calculate and publish as much as we 



350 LOGIC OF CHANCE. [CHAP. XV. 

please about the fate of any of the depressed classes 
at the bottom of the social scale, without any serious 
anxiety that our pi*edictions and conclusions would in 
consequence be &Jsified. But we should soon find the 
difference if we were to begin to discuss in this way 
the prospects of any persons who were likely to take 
an interest in our proceedings. It may be true that 
at present but little effect would be produced by any 
statements that we might publish about the future of 
society, because the possibility of making such state- 
ments is doubted; but if Sociology were ever to estab- 
lish its claims, the effects produced in each case by its 
own disturbing agency would rise into real importance. 
§ 8. The foregoing remarks apply principally to 
the case of a prediction being distinctly falsified owing 
to its statements being of a disagreeable character, but 
it must not be supposed that the difficulty is confined 
to such cases as these. The prediction may be equally 
falsified if it holds out an attractive prospect. There 
will not indeed be the same direct contradiction here, 
owing to the agents abstaining from what was foretold, 
but if, in consequence of the announcement, they per- 
form their actions more speedily or more effectually 
than they would otherwise have done, the prediction 
is still rendered incorrect. The conduct, as it is finally 
carried out, is not the consequence of the motives which 
had been taken into account only, but of these together 
with the additional motives suggested by the pub- 



SECT. 9.] LOGIC OF CHANCE. 351 

lication. The nature of the disturbance which would 
be thus produced, as dependent upon the character of 
the announcement made and the circumstances under 
which it was published, have never, I think, been dis- 
cussed, but seem well worthy of examination. 

An instance has already occurred (Ch. vii. § 18) of 
the kind of practical contradiction which might arise 
from the cause now under discussion. The subject is 
&r too wide for us to enter more fully into it here; 
I will only therefore call the reader's attention to the 
&ct of the existence of this cause of disturbance and 
the consequent caution that any inferences from our 
statistics cannot be warranted, when we extend them 
into the future, unless under the condition that the 
persons whose conduct is described either are left in 
ignorance of the statistics, or, if they know them, are 
still uninfluenced by them. 

§ 9. (II.) The remarks in the last few sections 
are intended to point out that that purely speculatiye 
and isolated position of the observer which alone ia 
tenable when we are laying down rules for a science 
of inference, is one which it is in certain cases prac- 
tically impossible to maintain. With every wish to 
be observers only we cannot always secure our isola- 
tion when we are describing the conduct of intelligent 
human beings, for we cannot always prevent them from 
being influenced by what we say. The criticism I 
have next to offer is of a veiy different kind. It refers 



352 WQIO OP CHANCE. [cHAP, XV. 

not to the actual disturbance caused unintentionally 
by the observer's published inferences, bat to an iu- 
tentional hypothetical disturbance in the actions which 
form the subject of the inference. The possibility of 
such a disturbance being contemplated arises from the 
fact that the observer himself, or other persons besides 
those to whom the inference refers, are themselves 
capable of acting in the same way as the persons 
whose conduct is described. Hence arise constant in- 
trusions of the observer's personality into calculations 
from which they should be rigidly excluded. The 
point may seem somewhat subtle, and 1 must therefore 
bespeak the reader^s attention to the following re- 
marks. 

§ 10. The statistics with which we are concerned 
in Probability are composed, as already stated, in great 
part of the voluntary actions of men. They may relate, 
for example, to crimes, such as the commonly adduced 
iustances of murders, thefts, and suicides; to virtuous 
actions, such as the sums annually expended in charit- 
able purposes; or to actions of an indifferent character, 
such as the number of marriages, or of insurances effected 
in the year. But of this portion of human conduct, 
as of most other portions, it is no matter of hypothesis, 
but a simple datum of expeiience, that in the long ran, 
when we extend our observations over a sufficient space, 
a great degree of uniformity is generally observabla 

§ 11. Now between statistics of this kind and 



1 



8BCT. 11.] liOGIC OF CSIAKCB. 353 

those which are concerned with what are not the im- 
mediate results of Yoluntary agency, whether the 
latter be of a purely involimtaiy character, as for ex- 
ample shipwrecks, or be resnlts in which the human 
will is generally but a remote cause, as throws of dice, 
or births and deaths, there is one marked difference. 
It is this. We the observers, or any one else whom 
we suppose to occupy the position of observer, are 
ourselves beings like those whose conduct we tabulate 
and reason about, and the actions in question are such 
as we are or may be in the habit of performing our- 
selves. Hence it restdts that we are conceivably, if 
one may so say, a portion of our own statistics; we 
may suppose our own case to be included in the 
statistics under discussion. In many of the common 
examples taken from insurance, and above all in games 
of chance, the case is of course extremely different. 
There we may preserve with perfect consistency that 
purely Phenomenalist view in which we regard our- 
selves as looking passively on the successions of exist- 
ences independent of ourselves. It would in feet be 
always difficult and often impossible not to take such 
a view there. 

But though not impossible, it is exceedingly diffi- 
cult to do the same when the things whose statistics 
we discuss are actions which men exactly like our- 
selves do perform, and which we any day may perform. 
To retain the correct view with rigid consistency it 

23 



354 IiOOIC OF CHANCE. [CHAP. XY. 

would indeed be necessary to exclude ourselves entirely 
from the statistics, in other words to confine ourselves 
consistently to the observer's point of view, as we 
unavoidably do in the case of games of chance. We 
might help to compose the statistics of others, just as 
others compose the statistics for us, but we must not 
attempt to occupy both positions, those of observer 
and observed, simultaneously. 

§ 12. I admit that owing to the peculiar chaiuc- 
ter of the series of statistics of Probability, and the 
merely a/verage truth with which we are there con- 
cerned, the inconsistent attempt just mentioned does 
not necessarily cause any error there. If indeed we were 
concerned with the absolute and universal statements 
of ordinary inference there would be error; the deter- 
mination of a man, for example, to commit suicide 
when the inferential statement in which he was in- 
cluded had contemplated his abstaining from such an 
act, would falsify the inference. But no one man has 
power, by his own private conduct at least, to do 
much injury, to an average. A registrar of deaths 
might drown himself as safely as any one else might, 
so far as affects the integrity of the formulse with which 
he is concerned. His conduct is an isolated event, 
whereas the statistics continue indefinitely. Hence 
although his suicide is formally unwarranted, owing 
to the fact of its being an intrusion into statistics 
which had not contemplated it, it soon becomes over- 



SECT. 14.] LOGIC OF CHANCE. 355 

rilled, and its effects, even had they ever been per- 
ceptible, gradually vanish from sight. In the long 
run such a disturbance of course could not show itself, 
whilst in the individual instance, by one of the funda- 
mental hypotheses of Probability (the irregularity of 
the details), we should not be justified in taking notice 
of the disturbance. 

§ 13. It is not therefore exactly by this stepping 
down of the obsei-ver into the arena of the statistics, 
unwarranted as it is, that the fallacy now to be 
noticed arises. It is rather by certain hypothetical 
intrusions to which the acknowledged practical harm- 
lessness of the actual intrusion gives rise, that error 
and confusion are caused. Finding that any one 
observer may without mischief do very much as he 
likes amongst the statistics, similar invasions are con- 
ceived upon such a scale as to involve the destruction 
of the speculative or scientific view, and, as we shall 
presently see, to caus^ amongst other things the ex- 
pression of a great deal of practical fatalism. 

§ 14. A quotation from 'Buckleys History of Civil- 
iaatian (Vol. i. p. 25) will form a convenient introduc- 
tion to the discussion now to be entered upon. After 
pointing out that among public and registered crimes 
there is none which seems so completely dependent on 
the individual, and so little liable to interruption as 
suicide, he proceeds as follows : — " These being the 
peculiarities of this singular crime, it is surely an 

23—2 



356 LOGIC OF CHANCE. [cHAP. XV. 

astonishing fact, that all the evidence we possess re- 
specting it points to one great conclusion, and can 
leave no doubt on our minds that suicide is merely 
the product of the general condition of society, and 
that the individual felon only carries into effect what 
is a necessary consequence of preceding circumstances. 
In a given state of society a certain number of persons 
must put an end to their own life*. This is the 
general law, and the special question as to who shall 
commit the crime depends of course upon special laws ; 
which however, in their total action, must obey the 
large social law to which they are all subordinate. 
And the power of the larger law is so irresistible, that 
neither the love of life nor the fear of another world 
can avail anything towards even checking its oper- 
ation." 

§ 15. The above passage as it stands seems very 
absurd, and would I think, taken by itself, convey an 
extremely unfidr opinion of its author's ability. But 
the views which it expresses are very prevalent^ ajid 
are probably increasing with the spread of statistical 
information and study. They have moreover a still 
wider extension in the form of a vague sentiment than 
in that of a distinct doctrine. And as they are not 
likely to find a more intelligent and widely read ex- 
positor, or to be expressed in a more vigorous and 
outspoken way, I do not think I can do better than 

'About 250 annually in London* 



gBCr. 17.] LOGIC OP CHANCE. 357 

state my opinions in the form of a criticism upon 
this quotation. 

§ 1 6. One portion of the quotation is plain enough. 
It simply asserts a statistical fact of the kind already 
familiar to us, namely, that about 250 persons annually 
commit suicide in London. This is all that the sta- 
tistics themselves establish. But, secondly, this datum 
of experience is extended by Induction. The inference 
is drawn that about the same number of persons will 
continue for the future to commit suicide. Now this, 
though not lying within the strict ground of the 
science of Probability, is nevertheless a perfectly legi- 
timate employment of Induction. The conclusion may 
or may not be correct as a matter of &.ct, but there 
can be no question that we are at liberty to extend 
our inferences beyond the strict ground of experience, 
and that the rules of inductive philosophy will fiimish 
us with many directions for that purpose. We may 
admit therefore that, for some time to come, the annual 
number of suicides will in all likelihood continue to be 
about 250. 

§ 17. But it will not take much trouble to show 
that there is a serious fallacy involved in most cases 
in the expression of such sentiments as those quoted. 
I am anxious that it should be clearly understood that 
this fallacy finds no countenance in either of the two 
assumptions which are necessary for the establishment 
respectively of the rules of Probability and Induction, 



358 LOGIC OF CHAKCB. [CHAF. XV. 

in those, namely, of statistical uniformity, and inyaria- 
bility of antecedence and sequence. In other words, 
the inference in the quotation would remain either un- 
meaning or &lse, in spite of our admitting that the 
number of persons who perform any assigned kind of 
action remains year by year about the same, and that 
the actions of each person are links in an inyari* 
able sequence*. 

§ 18. • We here have forced upon our attention the 
distinction between the two standing-points which may 
be occupied when we are investigating human conduct. 
Let us briefly examine them in turn. 

(1) There is, firstly, that speculative point of view 
which, as I have said, we are in consistency bound to 
retain. On this view all these asseiiions about the in- 
utility of efforts and the inefficiency of molives are 
meaningless, or rather inappropriate. "What we are 
then discussing is, not what people might do if they 
were to resolve to alter their conduct, but what we 

* It may prevent confosion if I remark here that my own 
opinion is in favour of Necessity, of the doctrine that w that 
where the antecedents are the same so will be the consequents, 
though I do not wish to speak with too great confidence. As 
stated above, such a doctrine is necessary for the establishment 
of strict rules of Induction, though not for that of Probability. 
The remarks in the last chapter were intended to show that 
whilst mere statistical uniformity scarcely had any bearing upon 
the question, it would not be easy by statistics of any kind 
rigidly to prove Necessity, and that when proved the doctrine 
would be comparatively barren of results as far as inferences are 
concerned. 



SECT. 19.] LOGIC OF CHANCE. 359 

have reason to infer that they loUl do. We are not 
concerned with the results of hypothetical alterations — 
these results might be of extreme impoi*tance — but 
we are drawing inferences as to whether such altera- 
tions will be made. All therefore that can be estab- 
lished by the fact of the statistical results remaining 
nearly the same is, that the amount of the counter- 
acting efforts and the strength of the antagonist motives 
remain the same, not that these efforts and motives are 
in any sense ineffective. To prove this last point it 
would be necessary to take very different ground, 
namely, to examine instances in which such efforts had 
been made and instances in which they had not, and 
to show that the results in each case were nearly or 
exactly the same. 

§ 19. (2) In distinction from the above there 
is the view taken from the practical standing-point. 
Every agent, whether or not his conduct form part 
of any table of statistics, finds himself the centre of a 
sphere of action. This view receives immense extension 
by each person being able to put himself in imagination 
into the position of any other individual, or into 
that of any body of individuals. When this position 
is occupied the question becomes a very different 
one from that last considered. We are not now con- 
sidering whether efforts loill be made, but we are dis- 
tinctly taking into discussion the different results ac- 
cording as they are made or not. This would be the 



360 LOGIC OF CHANCE. [CHAP. XT. 

most natural and appropriate explanation to be given 
of such remarks as those in the quotation before ua, 
and could be the only one offered if we were referring 
to the efforts of a single individual or to those of a 
few people. All that any person could then mean by 
talking about the inutility of efforts would have no 
reference to the question whether efforts would really 
be made or not; he would simply mean that the differ- 
ence, according as they were made or not, would be 
little or nothing. 

It will scarcely be maintained, in this sense, that 
motives are feeble or efforts at suppression ineffective. 
Any considerable alteration in the belief of people as to 
a future world, or in their comfort in this world, would 
unquestionably have a great influence upon the num- 
ber of murders or suicides. As regards the efforts of 
the clergy or magistrates to suppress the evil, however 
much these may be depreciated, it will not I apprehend 
be denied that a great deal might be done towards 
mcreasmg the annual number of those who destroy 
themselves, — ^by removing the poHce, for instance, from 
the neighbourhood of the Serpentine and Waterloo 
Bridge. And it tells equally for our present argu- 
ment, if it be admitted that the efforts of such per- 
sons could produce any consequence whatever, whether 
favourable or adverse. 

§ 20. The reply to this would probably be, that 
though considerable consequences might really follow 



SECT. 21.] LOGIC OF CHANGE. 361 

^were we to suppose an alteration in the conduct of 
persons in authority, or in the belief of the people, 
yet that we have no right to introduce such an ima- 
ginary alteration, because we know that as a matter 
of fiwjt it will not really take place. This is probably 
quite true, and I have no intention of denying it; but 
what I wish to draw attention to, and what seems to 
be often overlooked, is how in such a reply as this we 
may be shifting from one point of view to another. We 
are abandoning the view taken in the last section 
and falling back upon the speculative view. When 
the efforts of a few persons are contemplated, the 
hypothesis of their acting otherwise is admitted, but 
the consequent effect is pronounced to be insignificant, 
as might very likely be the case. When however the 
efforts of many are contemplated, the hypothesis of 
their actiug otherwise and the consequent effect, which 
would then be great, are not admitted, on the plea that 
they are inconsistent with fact 

§ 21. Such a confusion as that discussed above 
may seem absurd, but I cannot help thinking that in 
this way considerable support is often given to that 
practical fatalism which expresses itself in the common 
complaints about the utter impotence of the individual, 
and the irresistible power of great social laws, and 
which shows itself in our conduct by a selfish disposi- 
tion to let everything good or evil take its own coarse 
without troubling ourselves about it. It is observed 



362 LOGIC OF CHANCE. [CHAP. XV, 

that in the statistics of actions which may be the 
result^ in their final form, of many different motives 
and of various conflicting struggles, there is yet year 
by year a marked uniformity. Instead of concluding 
from this, what alone ought to be concluded from 
the standing-point of a science of inference, that the 
motives and the efforts remain about the same year 
by year, a conftision is made between this and the 
practical view, and the doctrine is obtained that the 
efforts at repressing such conduct are unavailing. 

§ 22. Such fatalistic views are often expressed in 
the form of disparaging comments upon the insignifi- 
cance of individual efforts. In the sense in which this 
complaint is often made, I cannot but think that it is 
nothing more than an expression of our own selfishness, 
and really means, not that the results we could effect 
are small, but that we care little about them. Let us 
test this by taking some statistical imiformity, in which 
the motives that act to produce the result are almost 
entirely of a self-regarding character. Suppose that 
any one having ascertained that about a thousand per* 
sons, daily, in London, who could afford to eat a dinner, 
do for one reason or another go without it, were to 
announce this fact as a great social law of prodigious 
efficacy, and were to declare that its power was such, 
when examined on a large scale, that neither the fear 
of future hunger nor the love of good food could pre- 
vail towards even checking its operation, what would be 



SECT. 22.] LOGIC OP CHANCE. 363 

the natural reply? The form of these statistics and the 
nature of the argument grounded upon them are 
identical with those of the example cited by Mr 
Buckle. The reply would probably be, that if we were 
professing simply to draw inferences, most of this talk 
about the impossibility of checking such actions was, to 
say the least, inappropriate ; but that if we were taking 
the practical view, that is, if we were for any purpose 
putting ourselves into the position of one or more of 
the individuals in question, the statement was utterly 
false. Each one of those men could in most cases 
have eaten his dinner or not according as he pleased, 
and therefore the whole body could have done so. 
And no sophistry about free-will and necessity would 
be allowed for a moment to stand in the way of such 
a judgment Equally absurd would it be considered 
to talk about the insignificance of individual efforts 
in the face of such a great social law. But were people 
perfectly unselfish, statistics of crimes would not 
differ much from such an example as this. Almost 
any one individually might do something, and small 
bodies of men might do a great deal, towards diminish- 
ing crime, were it but in a single instance. When 
therefore it is vaguely complained that efforts are 
fruitless in the case of crimes, is there not some ground 
to think that the real meaning is that such efforts, on 
any much larger scale, are not likely to be made ? And 
when it is urged that the individual can do nothing 



364 LOGIO OF CHA17CB. [CHAP. XV. 

in this case, whilst no one would dream of making 
the same assertion in the other case, are we wrong in 
thinking that the real difference is that the attain- 
ment of one's own dinner is more universally regarded 
as a substantial good than the suppression or diminu- 
tion of our neighbour's faults] I do not, of course, 
deny that we should find it much more easy to dissuade 
people from some courses of conduct than from others ; 
all that I mean is, that there is no real distinction 
between the general deductions which may be drawn 
from one or another kind of statistical regularity. . 

§ 23. A few remarks, chiefly by way of summary, 
may be offered in conclusion, inadequate as they are for 
the discussion of so important a subject as that treated 
of in this chapter. 

We have been engaged, throughout this Essay, in 
considering the laws of inference applicable to the 
class of things which combine individual irregularity 
with aggregate regularity. Human actions of most 
kinds possess this property as unquestionably as do 
the throws of a die or the casualties caused by storm& 
The same laws of inference therefore which apply to 
the latter kind of events apply also to the former. 
But in saying this we are no more putting these two 
kinds of events upon the same footing than the histo- 
rian is neglecting the distinction between virtue and 
vice when he employs the same rules of arithmetic 
to^ reckon up, say, the numbers who in any country 



SECT. 23.] LCX3IC OF CHAXCE. 865 

have been burnt at the stake as martyrs and hanged 
on the gallows as thieves. To say that our attention 
should, for purposes of inference only, be fixed upon 
one quality in an event, does not imply that we are 
to forget the existence of other qualities in that event. 
When we come to examine the actions to which the 
statistics refer, we find of course that they possess 
many other properties besides that which makes them 
fitting subjects of Probability. Their consequences 
may be of overwhelming importance, they may be the 
result of long deliberation and of bitter mental conflict, 
they may be morally admirable or detestable. But 
with all these latter qualities the logician, as such, is 
not concerned. His task is to make inferences. He 
has to calculate the chance of an event, whether 
that event be the throw of a die, a shipwreck, or a 
siiicide. 

Select almost any kind of action, and we shall 
find, if we extend our observation over a sufficiently 
large body of men, that there is a uniformity in the 
performance of that action. I have already called 
attention to some erroneous inferences which are often 
drawn from the existence of such uniformity. It was 
pointed out in the last chapter, that causation, in 
the individual instances, could not be proved in this 
way. It has also been repeatedly insisted on that to 
generalize as to the continuance of such uniformities 
beyond the limits within which they have been ac- 



366 LOGIC OF CHANCE, [CHAP. XV, 

tually observed, though in many cases perfectly legiti- 
mate, belongs to Induction and not to Probability. 

But at present we are not concerned with such 
considerations as these. We will assume that there 
is a long-continued uniformity in the frequency of the 
performance of some action, against which, it may be, 
large classes of persons are struggling with their whole 
strength. What we are now concerned with is the 
vital importance of the distinction between what may 
be called the speculatiye and the practical views which 
we may take in reference to any such uniformity. 

What we have to adhere to, in making inferences, 
is the speculative view. On this view we have no 
right, when talking about the future, to use any other 
expressions than those which denote simple futurity. 
To say that the agents *must' perform certain actions, 
or 'cannot' perform others, is inadmissible. To say 
this is to fall into a* fatalistic fallacy, for it generally 
involves a confusion between certainty of inference 
on our own part and oomptdsion on the part of the 
agents. 

But against fatalism, in anything which has a 
close connexion with our own comfort or convenience, 

* By Fatalism I understand the doctrine that events really 
dependent in part upon human agency, will yet be equally 
brought to pass whether men try to oppose or to forward them. 
It is essentially distinct from Necessity, and is indeed rather 
a sentiment than a doctrine, which it is difficult to state with 
brevity without making it obviously involve a contradiction* 



SECT.. 23.] LOGIC OF CHANGE. 367 

the ordinary European mind is protected by a healthy 
instinct of incredulity. We should try in vain, by 
any effort, to persuade people that each agent could 
not generally alter his conduct if he pleased, or, con- 
sequently, that any body of men could not produce 
an appreciable effect if they were to try. The plain 
man feels that such statements as these are absurd; 
the thinker knows that, whichever way the doctrine 
of Necessity be settled, that doctrine does in no way 
whatever come into contact with the practical problems 
of life when stated in the above form. 

When however the efforts of the agent are directed 
towards securing, not his own good but the good of 
others, the promptings of our natural indolence and 
selfishness offer dangerous facilities for the reception 
of such fatalistic doctrines, at least in the minds of 
those who are only looking on at the struggle and not 
sharing in it themselves. It is one thing to entertain 
the conviction that certain results will remain for 
some time uniform, because the conflicting efforts 
which produce them remain uniform. It is quite 
another thing to conclude that the efforts on one side 
are themselves ineffective. But the mere spectator, 
if his sympathies are not active, flnds it only too easy 
to step across the logical chasm which sepai*ates one 
of these opinions from the other. I cannot help think- 
ing that much support is thus given to the doctrine 
which one hears uttered in so many different forms. 



368 LOGIC OF CHAHCS. [CHAP. XT. 

and in every shade of dogmatism, by a certain school 
of writers, that the sorrows and the crimes of our 
fellow men are only the necessary product of the 
existing state of society, and that the efforts of the 
individual are insignificant. Thei'e are many perhaps 
who would have indolently told a Howard or a 
Wilberforce that he could do nothing, who would yet 
be very much astonished if asked whether the trouble 
of their own doctor in coming to see them produced 
insignificant results. 

But the confusion between the speculative and 
the practical points of view produces, I think, still 
farther consequences, quite as deplorable as those 
already described. Neither the Necessitarian nor 
the Fatalist need be men of blunted moral feelinga 
We might, that is, hate a wrong action though we 
thought it inevitable in the sense that the agent 
would not as a matter of fact choose to avoid com- 
mitting it ; we might even hate such an action though 
we thought it inevitable in the sense that the agent 
must do it whether he chose or not But the com- 
plaint is often made, and I think not altogether un- 
justly, that the advocates of Sociology are too much in 
the habit of regarding crimes as being not only cer- 
tain to happen, but as being morally indifferent 
In so far as this complaint is true, I should think 
that such an apparent moral obtuseness of judg- 
ment (I shall not be misunderstood as hinting that 



SBCT. 23.] LOGIC OF GHAHG& 369 

this is accompaiiied by moral ludtyin pnctioe) is con- 
nected with that oonfosion between two distinct views 
'^liidi bas occapied our attention during this chap- 
ter. Tbe connection woold be as follows. The 
specolatiye view is in one sense wider than the prac- 
ticaly for tbe former inclndes not only Tolontary actions, 
(tbe province of the practical view,) bat also actions 
^vbicb are not voluntary, as well as results wbich are 
not strictly speaking actions at all, such as the &ces 
turned up by dica In the great majority of sub- 
jects to whicb this view introduces us, moral praise 
and blame have no applicability. When therefore the 
two views are confused together, we are sometimes apt, 
not merely to hamper our practice by &talism, but 
even to run the risk of debasing our moral judgment 
by regarding the actions of men with the indifference 
witb wbich we regard the happening of things. There 
is danger, for example, lest we should not merely be- 
lieve that the Jiumber of murders or suicides are so 
fixed that efforts are unavailing to counteract them, 
but even that we should feel little more affected at the 
commission of crimes than at tbe successions of the 
throws of a die. 

Against every sucb confusion between two views 
there is no safeguard comparable with that afford- 
ed by the habit of familiarizing ourselves witb each 
view. In other words let us temper our speculations 
with a wholesome infusion of practice. Fatalism can- 

24 



370 liOGIC OP C?HAKCE. [CHAP. XV. 

not easily exist in the fresh air of practical life. The 
hardest workers are generally the most hopefdl men, 
and in our own unselfish efforts will be found the bedt 
corrective to that depression which is so apt to be pro- 
duced by a too exclusive devotion to the speculative 
view. We shall thus avoid the danger of always dis- 
cussing the joys and the sorrows of our fellow-men in a 
way which, though legitimate when we are avowedly 
taking a partial view of the subject, too easily lapses 
into hopeless indifference or cynicism if we suffer our- 
selves to forget how partial that view is. 



THE END. 



CAMBBIDCfE: PBINTBD AT THE UNIVBBSITr FBSSflL 



^s